Semester 2 Problems#
Solutions are available for you to check your answers.
Chapter 6#
Differentiation from first principles#
On a careful plot of \(y = x^2\), draw the tangent to to the curve at \((-2,4)\) and find its gradient. Check your result against the formula from lectures for differentiating powers.
(The aim of this question is to make you realise how difficult it is to get correct answer using this method.)
(i) Let \(f(x)=8x^2 + 2\). By differentiating \(f(x)\) using first principles, find the gradient of \(y=8x^2 + 2\) at a typical point \((x,8x^2 + 2)\).
(ii) Now let \(f(x)=8x^2 + 4\). Differentiate \(f(x)\) using first principles and compare your answer with part (i).
(iii) Without doing any more workings, what would the result be if you started with \(f(x)=8x^2+6\)?
Differentiate \(f(x) = 4x^2 -4x + 4\) from first principles; that is, find the gradient of \(y = 4x^2 -4x + 4\) at a typical point \((x, 4x^2-4x+4)\).
(i) Let \( f(x) = \frac {1}{x}\). By working over a common denominator, simplify \(f(x+h)-f(x)\). Hence differentiate \(f(x)\) from first principles.
(ii) Let \( g(x) = \frac{1}{x^2}\). By working over a common denominator, simplify \(g(x+h)-g(x)\). Hence differentiate \(g(x)\) from first principles.
Differentiating powers#
Find the derivatives of
(i) \(f(x) = x^{-2}\),
(ii) \(f(x) = \sqrt{x}\),
(iii) \(f(x) = x^{-\frac 34}\).
Let \(\displaystyle y = \frac{4x^3}{3x^{\frac 12}}\). Find \(\frac{dy}{dx}\).
Let \(\displaystyle y = \frac{2x^2 + x}{\sqrt{x}}\). Find \(\frac{dy}{dx}\).
Differentiate each of the following functions.
(i) \(f(x) = 2x(3x^2-4)\),
(ii) \(g(x)=\displaystyle\frac{10x^5+3x^4}{2x^2}\),
(iii) \(h(t)=(t+1)(t-2)\),
(iv) \(k(s)=\displaystyle\frac{2s^3-s^2}{3s}\).
Tangents and normals#
Let \(y = 2x^2 -3x - \frac 1x\). Find the equations of the tangent and normal to the curve at the points \(P= (1,-3)\) and \(Q= (-1,6).\)
Find the equations of the tangents \(t(x)\) and the normals \(n(x)\) to thefollowing curves at the points corresponding to the given values of \(x\):
(i) \(y=x^2\), \(x=2\)
(ii) \(y=3x^2 +2\),\(x = 4\).
Small changes formula#
The surface area of a sphere is \(4 \pi r^2\), where \(r\) is the radius. If the radius of the sphere is increased from 10cm to 10.1cm, what is the approximate increase in surface area?
The height of a cylinder is 10cm and its radius is 4cm. Find the approximate increase in volume when the radius increases to 4.02cm.
When measuring the area of a circle, 2% error is made. Find the percentage error in the radius.
Chapter 7#
Stationary points#
Find the stationary points of \(y = f(x) = 3x -x^2\) and determine their nature using Method I from the notes.
Investigate the stationary values of \(y\) on the following curves using Method I.
(i) \(y=x^4\),
(ii) \(y=3-x^3\),
(iii) \(y=x^3(2-x)\),
(iv) \(y=3x^4 +16x^3 +24x^2 +3\).
Let \(y = x^4 -2x^3 +7x^{-1} + x^{\frac 12}\) and find \(\frac{d^2y}{dx^2}\) and \(\frac{d^3y}{dx^3}\).
Let, in turn, \(y = x\), \(y = x^2\), \(y = x^3\), \(y = x^4\), \(y = x^5\). Find in each case \(\frac{dy}{dx}\), \(\frac{d^2y}{dx^2}\), \(\frac{d^3y}{dx^3}\), \(\frac{d^4y}{dx^4}\), \(\frac{d^5y}{dx^4}\) and \(\frac{d^6y}{dx^6}\).
Hence, for a positive integer \(n\), find a formula for \(\frac{d^2}{dx^2}\left(x^n\right)\), \(\frac{d^3}{dx^3}\left(x^n\right)\), \dots, \(\frac{d^{n+1}}{dx^{n+1}}\left(x^n\right)\).
Find the stationary points of \(y = x^4 - \frac 83 x^3 + 2x^2 +1\) and determine their nature using the second method when possible.
Find the turning points of the following curves and determine their nature using the method of second derivatives when possible:
(i) \(y=x(x^2-12)\),
(ii) \(y=x(x-8)(x-15)\),
(iii) \(y=t^2(3-t)\),
(iv) \(y=\displaystyle 4x^2 +\frac 1x\).
Graph sketching using stationary points#
Sketch \(y = x^4 - 8x^2 +7.\)
Sketch the graphs of the following functions:
(i) \(y=x^3 -2x^2 +x\),
(ii) \(y= x^3 -2x^2 +x\),
(iii) \(y = (x+1)^2(2-x)\),
(iv) \(y=x^2 (x-2)^2\),
(v) \(y=x^4 -8x^3\).
Sketch \(\displaystyle y = f(x) = 3x - \frac{12}{x}\).
Sketch the graph of the following functions:
(i) \(y=\displaystyle \frac 1{x^2}\),
(ii) \(y=\displaystyle \frac 1{x^3}\),
(iii) \(y=\displaystyle \frac 1{x^4}\),
(iv) \(y=\displaystyle x - \frac 1x\),
(v) \(y=\displaystyle x + \frac 4{x^2}\),
(vi) \(y=\displaystyle x - \frac 4{x^2}\).
Applications to optimisation#
A solid rectangular block has square base. Find its maximum volume if the sum of the height and any one side of the base is 12cm.
A ball is thrown vertically upwards from ground level and its height after \(t\) seconds is \((15.4t -4.9t^2)\)m. Find the greatest height it reaches, and the time it takes to get there.
Consider a rectangular sheet of metal of length \(a~\mbox{cm}\) and width \(b~\mbox{cm}\). Equal squares of side length \(x~\mbox{cm}\) are removed from each corner, and the edges are then turned upwards to make an open box of volume \(V~\mbox{cm}^3\). Find a formula for \(V\). Hence find the maximum possible volume, and the corresponding value of \(x\), when
(i) \(a = 8\) and \(b = 5\),
(ii) When \(a\) and \(b\) are not further specified.
Chapter 8#
Chain rule#
Let \(y = (6x^3-4x)^{-2}\). Find \(\frac{dy}{dx}.\)
Differentiate the following.
(i) \(y=(2x+3)^2\),
(ii) \(y=2(3x+4)^4\),
(iii) \(y=(2x+5)^{-1}\),
(iv) \(y=\displaystyle (3x-1)^{\frac 23}\),
(v) \(y=\displaystyle (3-2x)^{-\frac 12}\),
(vi) \(y=(3-4x)^{-3}\),
(vii) \(y=\displaystyle \frac{1}{3x+2}\),
(viii) \(y=\displaystyle \frac 1{(2x+3)^2}\),
(ix) \(y=\displaystyle \frac 1{\sqrt{3x+1}}\),
(x) \(y=\displaystyle \frac 1{(2x-1)^\frac 23}\),
(xi) \(y=\displaystyle \frac{3}{\sqrt{2+x^2}}\),
(xii) \(y=\displaystyle (ax+b)^n\), (\(a\), \(b\), \(n\) constants).
Rates of change#
The area of the surface of a sphere is given by the formula \(4\pi r^2\), \(r\) being the radius. Find the rate of change of the surface area in \(\mbox{cm}^2\mbox{s}^{-1}\) when \(r = 2~\mbox{cm}\) given that the radius increases at the rate of \(1~\mbox{cm}\) per second.
The area of a circle is increasing at the rate of 3cm\({}^2\) per second. Find the rate of change in the circumference when the radius is 2cm.
If \(y = x^2-3x\) and \(x\) is a function of \(t\) such that \(\frac{dx}{dt} = 2\), find \(\frac{dy}{dt}\) when \(x=2\).
If \(y = (x -\frac 1x)^2\) and \(x\) is a function of \(t\), find \(\frac{dx}{dt}\) at \(x=2\) given that at the value of \(\frac{dy}{dt}\) at \(x=2\) is 1.
Product rule#
Differentiate
(i) \(f(x)= (x+1)(x+2)\),
(ii) \(f(x)=(x^2+1)x^2\),
(iii) \(f(x)=(x-2)^2(x^2-2)\),
(vi) \(f(x)=(x+1)^2(x^2-1)\),
(v) \(f(x)=x^3(5x+1)^2\),
(vi) \(f(x)=2\sqrt{x}(x+1)^2\),
(vii) \(f(x)=x^{-2}(1+3x)^2\),
(viii) \(f(x)=x^2(1+x)^{-2}\).
Quotient rule#
Find the first and second derivatives in each of the following.
(i) \(f(x)=\displaystyle \frac{x}{x+1}\),
(ii) \(f(x)=\displaystyle \frac{x}{x-1}\),
(iii) \(f(x)=\displaystyle \frac{1+5x}{5-x}\),
(iv) \(f(x)=\displaystyle \frac{x}{(x+3)^4}\).
Chapter 9#
Differentiating exponentials#
Differentiate \(\displaystyle f(t)= 2\sqrt{\frac t{x+t}}\), assuming \(x\) is kept constant.
Differentiate
(i) \(y = e^{2x^{-\frac{1}{2}}}\),
(ii) \(y = e^{x^2+x+1}\),
(iii) \(y = e^{(3x-1)^2}\).
Differentiate the following:
(i) \(f(x)=4e^x\),
(ii) \(f(x)=e^{3x}\),
(iii) \(f(x)=e^{2x+1}\),
(iv) \(f(x)=e^{2x^2}\),
(v) \(f(x)=x^2e^{x}\),
(vi) \(f(x)=\displaystyle \frac{e^x}{x}\),
(vii) \(f(x)=e^{x^2}\),
(viii) \(f(x)=e^{2x^3}\),
(ix) \(f(x)=e^{-x^2}\),
(x) \(f(x)=e^{\sqrt{x}}\).
Find the equation of the tangent to the curve \(y = e^x\) at the point \((a, e^a)\). Deduce the equation of the tangent to the curve which passes through the point \((0,1)\).
Differentiating logs#
Differentiate
(i) (i) \(y = \ln(2x-1)\),
(ii) \(y = \ln(3x^2 +x^\frac 12)\),
(iii) \(y = e^{2x}\ln(x)\),
(iv) \(\displaystyle y=\frac{\ln(x)}{e^{x^2}}\).
Differentiate the following with respect to \(x\).
(i) \(f(x)=\ln(x^2-2)\),
(ii) \(f(x)=\ln(x\sqrt{x+1})\),
(iii) \(f(x)=\ln(3x)\),
(iv) \(f(x)=\ln(4x)\),
(v) \(f(x)=\ln(ax)\) (\(a > 0\) constant),
(vi) \(f(x)=\ln(3x+1)\),
(vii) \(f(x)=\ln(2x^3)\),
(viii) \(f(x)=\ln(x^3-2)\),
(ix) \(f(x)=\ln(x-1)^3\),
(x) \(f(x)=4\ln(x)\),
(xi) \(f(x)=\displaystyle \ln\sqrt{\frac{1-x}{1+x}}\),
(xii) \(f(x)=\ln(x\sqrt{x^2+1})\),
(xiii) \(f(x)=\displaystyle \ln\left(\frac{(x+1)^2}{\sqrt{x-1}}\right)\).
Differentiate the following with respect to \(x\).
(i) \(f(x)=x\ln(x)\),
(ii) \(f(x)=x^2\ln(x)\),
(iii) \(f(x)=\displaystyle \frac{\ln(x)}x\),
(iv) \(f(x)=\displaystyle \frac{\ln(x)}{x^2}\),
(v) \(f(x)=\displaystyle \frac x{\ln(x)}\),
(vi) \(f(x)=(\ln(x))^2\),
(vii) \(f(x)=\ln(\ln(x))\),
(viii) \(f(x)=\ln(\ln(x)^k)\) where \(k \in \mathbb{R}\) is a constant,
(ix) \(f(x)=\ln(ax+b)\),
(x) \(f(x)= x\ln(t)\), where \(t\) is held constant.
Differentiate \(y = \displaystyle\ln\left(\frac{x^4(3x^2-1)^5}{(x^\frac 12 +2)^3}\right)\). [Hint: Use laws of logarithms to simplify the expression before starting.]
Chapter 10#
Differentiating standard trig functions#
Differentiate the following with respect to \(x\).
(i) \(f(x)=\sin(2x-3)\),
(ii) \(f(x)=\cos(3x -1)\),
(iii) \(f(x)=\cos 3x -1\),
(iv) \(f(x)=-3\cos 5x\),
(v) \(f(x)=-4 \sin\left(\frac 32x\right)\),
(vi) \(f(x)=2 \sin\left( \frac 12(x+1)\right)\),
(vii) \(f(x)=\sin^2(x)\),
(viii) \(f(x)=4\cos^2(x)\),
(ix) \(f(x)=\cos^3 x\),
(x) \(f(x)= 2 \sin^3 (x)\),
(xi) \(f(x)=3\cos^4 (x)\),
(xii) \(f(x)=\sqrt{\sin(x)}\),
(xiii) \(f(x)=\cos^2(3x)\),
(xiv) \(f(x)=-2\sin^3 (3x)\),
(xv) \(f(x)=3\sin^4 (2x)\),
(xvi) \(f(x)=\sqrt{\sin(2x)}\),
(xvii) \(f(x)=x^2\sin(x)\),
(xviii) \(f(x)=\sin(x) \cos(x)\),
(xix) \(f(x)=\displaystyle \frac {\sin(x)}x\),
(xx) \(f(x)=\displaystyle \frac{\cos(2x)}x\),
(xxi) \(f(x)=\displaystyle \frac x{\sin(x)}\),
(xxii) \(f(x)=\displaystyle \frac {x^2}{\cos(x)}\).
Differentiating reciprocal trig functions#
Differentiate the following with respect to \(x\).
(i) \(f(x)= \tan (2x)\),
(ii) \(f(x)=\cot (3x)\),
(iii) \(f(x)=3 \sec (2x)\),
(iv) \(f(x)=2 \text{ cosec}\left(\frac 12 x\right)\),
(v) \(f(x)=-\tan(2x+1)\),
(vi) \(f(x)=\frac 13 \sec(3x -2)\),
(vii) \(f(x)=-2\cot(3x+2)\),
(viii) \(f(x)=\cot\left(x^2\right)\),
(ix) \(f(x)=\tan\left(\sqrt{x}\right)\),
(x) \(f(x)=\tan^2(x)\),
(xi) \(f(x)=\sec^2(x)\),
(xii) \(f(x)=2 \cot^3 x\),
(xiii) \(f(x)=3\text{ cosec}^2(x)\),
(xiv) \(f(x)=-\tan^2(2x)\),
(xv) \(f(x)=\frac 12 \cot^2(3x)\),
(xvi) \(f(x)=x\tan(x)\),
(xvii) \(f(x)=\sec(x) \tan(x)\),
(xviii) \(f(x)=x^2\cot x\),
(xix) \(f(x)=3x \text{ cosec} x\),
(xx) \(f(x)=\text{ cosec} x \cot x\),
(xxi) \(f(x)=\displaystyle \frac{\tan(x)}x\),
(xxii) \(f(x)=\displaystyle \frac {\sec(x)}{x^2}\),
(xxiii) \(f(x)=\sin(x) - x \cos(x)\),
(xxiv) \(f(x)=x\sec^2(x) - \tan(x)\).
Differentiating inverse trig functions#
Differentiate the following with respect to \(x\).
(i) \(f(x)= \sin^{-1}(2x)\),
(ii) \(f(x)=\cos^{-1}(3x-4)\),
(iii) \(f(x)=\tan^{-1}(x^2)\),
(iv) \(f(x)=\sin^{-1}(x^4 +3)\),
(v) \(f(x)=\left(\cos^{-1}(x)\right)^2\),
(vi) \(f(x)=\tan^{-1}(e^x)\),
(vii) \(f(x)=(\tan^{-1}(x))^{-1}\),
(viii) \(f(x)=(\tan^{-1}(x))^2\).
Mixed examples#
In each case find \(f'(x)\).
(i) \(f(x)=\cos(\sin(x))\),
(ii) \(f(x)=\sin(\cos(x))\),
(iii) \(f(x)=e^{\cos(x)}\),
(iv) \(f(x)=e^{\sec(x)}\),
(v) \(f(x)=e^{3\tan(x)}\),
(vi) \(f(x)=e^{\sin(2x)}\),
(vii) \(f(x)=\frac x2 e^{\sin(x)}\),
(viii) \(f(x)=e^{x^2}\tan^{-1} x\),
(ix) \(f(x)=\ln\left(\sin^2(x)\right)\),
(x) \(f(x)=\ln\left(\sin(x^2)\right)\),
(xi) \(f(x)=\ln\left(\tan^{-1} \left(\frac x2\right)\right)\),
(xii) \(f(x)=\ln\left(\cos^{-1}(x)\right)\),
(xiii) \(f(x)=\ln\left(\frac{\sin(x) + \cos(x)}{\sin(x) - \cos(x)}\right)\),
(xiv) \(f(x)=\ln\left(3x\cos^2(2x)\right)\).
Chapter 11#
Parametric differentiation#
Let \(x = t-1\), \(y = 2t+1\) for \(t \in \mathbb{R}\). What is cartesian equation of this curve? Find \(\frac{dy}{dx}\).
Let \(x = \cos(t)\), \(y = \sin(t)\) for \(0 \leq t \leq 2\pi\). On a diagram plot the points corresponding to \(t=0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4},\frac{3\pi}{2}\) and \(\frac{7\pi}{4}\). What shape is formed?Find \(\frac{dy}{dx}\) in terms of \(t\).
Let \(a\), \(b\), \(c\) and \(d\) be real numbers with \(a \neq 0\) and \(c \neq 0\). Let \(x = at + b\) and \(y = ct + d\) for \(t \in \mathbb{R}\). By eliminating \(t\) show that the curve defined by this parametric equation is a straight line.
Remark 1
This result tells us that if both \(x\) and \(y\) are parametrised by linear polynomials, the resulting curve is a straight line.)
Let \(\displaystyle x = \frac{2t}{1+t^2}\) and \(\displaystyle y = \frac{1-t^2}{1+t^2}\) for \(t\in\mathbb{R}\). Find \(\frac{dy}{dx}\) in terms of \(t\).
If \(\displaystyle x = \frac{1}{\sqrt{1+t^2}}\) and \(\displaystyle y = \frac{t}{\sqrt{1+t^2}}\) find \(\frac{dy}{dx}\) in terms of \(t\).
Let \(\displaystyle x = \frac{t}{1-t}\) and \(\displaystyle y = \frac{1-2t}{1-t}\) for \(t\in \mathbb{R}\setminus\{1\}\). Find \(\frac{dy}{dx}\) in terms of \(t\).
Implicit differentiation#
Let \(x^2 +y^2 -6xy +3x -2y +5 = 0\). Find \(\frac{dy}{dx}\).
We define \(y\) implicitly as a function of \(x\) by \(x^2 +2xy + y^2 = 2\). Find \(\frac{dy}{dx}\). Do the points \((\sqrt{2},0)\) and \((0,-\sqrt{2})\) lie on the curve? If they do, find the gradient of the curve at these points.
Find the gradient of the ellipse \(2x^2 +3y^2 = 14\) at the points where \(x = 1\).
Find the \(x\)-coordinates of the stationary points of the curve represented by the equation \(x^3 -y^3 -4x^2 +3y = 11 x+4\).
Find the gradient of the tangent to the hyperbola \(xy = 6\) at the point \((2,3)\).
Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\) when \(x^2 +y^2 -2xy +3y -2x = 7.\)
Find \(\frac{dy}{dx}\):
(i) \(x^2y^3=8\),
(ii) \(xy(x-y)=4\),
(iii) \(3(x-y)^2=2xy+1\).
Differentiate the following:
(i) \(y=5^x\),
(ii) \(y=2^{x^2}\),
(iii) \(y= 7^{3x}\),
(iv) \(y=a^x\) for \(a>0\), \(a\neq 1\) fixed,
(v) \(y=3^{2x-1}\),
(vi) \(y=x2^x\),
(vii) \(y=10^x\),
(viii) \(y=a^{tx}\), where \(a > 0\),
(ix) \(y=x^{2x}\),
(x) \(y=x^{\sin(x)}\).
Chapter 12#
Indefinite integration#
Let \(f'(x) = x^4 +x^2 + x^ \frac 12 + x^{-2}\). Find \(f(x).\)
Find \(\displaystyle\int 7x^4 + \frac 13 x^2 + \frac{1}{x^\frac 23} dx\).
Integrate
(i) with respect to \(x\): \(\frac 12\), \(\frac 12 x^2\), \((2x+3)^2\), \(x^{-5}\), \(\displaystyle \frac{-2}{x^4}\);
(ii) with respect to \(t\): \(at\), \(\frac 13 t^3 + \pi\), \((t+1)(t-2)\), \(\displaystyle \frac{1}{t^{n+1}}\) for \(n \neq 0\), \(\displaystyle \frac 1{t^2} + 3 + 2t\);
(iii) with respect to \(y\): \(-ay^2\), \(\displaystyle \frac k{y^2}\), \(\displaystyle\frac{(y^2+2)(y^2-3)}{y^2}\).
For each of the following, find the equation of the curve \(f(x)\) with the given \(f'(x)\) and which goes through the given point:
(i) \(f'(x) = 3x^2 -2x\) and \(f(1) = 1\);
(ii) \(f'(x) = \displaystyle-\frac 76 x^{-\frac{13}{6}} + 6x + 8x^{-3}\) and \(f(1) = 2\);
(iii) \(f'(x) =\displaystyle \frac 1{2\sqrt{x}} - 2\pi x\) and \(f(\pi^2) = \pi(1-\pi^4)\);
(iv) \(f'(x) =\displaystyle \frac{\ln 2}{3}x^{-\frac 23} + \frac 5{x^2} + 14 x^5\) and \(f(1) = \ln 2 - \frac 83 + e\).
Use a table of standard derivatives to calculate the following indefinite integrals:
(i) \(\displaystyle\int 3\cos(x)dx\),
(ii) \(\displaystyle\int\left(x^2+\sin(x)+2\cos(x)\right)dx\),
(iii) \(\displaystyle\int\sec^2(x)dx\),
(iv) \(\displaystyle\int e^x dx\),
(v) \(\displaystyle\int \frac{1}{x^2+1} dx\),
(vi) \(\displaystyle\int\frac{1}{x}dx\),
(vii) \(\displaystyle\int\text{ cosec}(x)\cot(x)\),
(viii) \(\displaystyle\int\frac{1}{\sqrt{1-x^2}}dx \; \text{ where } x<1\).
Area and definite integration#
Find the area \(S\) under the graph of \(y = x^2\) lying over the following regions of \(x\)-axis:
(i) \(3\) to \(6\),
(ii) \(a\) to \(b\),
(iii) \(0\) to \(x\)
Evaluate
(i) \(\displaystyle\left[ \frac {x^4}{4}\right]_{\frac{1}{2}}^2\),
(ii) \(\displaystyle\Big[3x^3 -4x\Big]_{-1}^1\),
(iii) \(\displaystyle\left[\frac 16x^3- 3x^2 + \frac 12 x\right]_{-2}^{-1}\),
(iv) \(\displaystyle\left[x^3 -\frac 1{x^2}\right]_{-4}^{-3}\).
Find the area enclosed by \(x + 4y - 20 = 0\) and the axes, by integration.
Find the areas enclosed by the \(x\)-axis, and the following curves and straight lines:
(i) \(y = x^2 + 2\), \( x= -2\), \(x= 5\);
(ii) \(y = x^2(x-1)(x-2)\), \( x= -2\), \(x= -1\);
(iii) \(y = \frac{3}{x^2}\), \( x= 1\), \(x= 6\).
Find the area under \(y = 4x^3 + 8x^2\) from \(x = -2\) to \(x =0\).
Sketch the curve \(y = f(x) = x^2 - 5x +6\) and find the area cut off below the \(x\)-axis.
Sketch the following curves and find the areas enclosed by them, and by the \(x\)-axis, and the given straight lines:
(i) \(y = -x^3\), \(x = -2\);
(ii) \(y=\frac{1}{x^2}-1\), \(x = 2\).
Find the area of the segment cut off from \(y = x^2 - 4x + 6\) by the line \(y = 3\).
Find the point(s) of intersection of the curve \(y = (x+1)(x-2)\) and the line \(x-y+1 = 0\), and calculate the area of the enclosed by these graphs.
Find
(i) \(\displaystyle \int_{\ln 2}^{\ln 3} 2e^xdx\);
(ii) \(\int_1^e \frac 1xdx\);
(iii) \(\displaystyle\int_0^{\ln\sqrt{2}} e^{-2x}\).
Find
(i) \(\displaystyle \int_0^{\frac \pi2} \cos(x)dx\);
(ii) \(\displaystyle\int_0^{\pi} \sin(x)dx\);
(iii) \(\displaystyle\int_{-\frac \pi 2}^{\frac \pi 2} \cos(x)dx\).
Find
(i) \(\displaystyle\int_0^{\frac{\pi}{4}}\sec^2(x)dx\),
(ii) \(\int_0^{\frac{\sqrt{3}}2} \frac 1{\sqrt{1-x^2}}dx\),
(iii) \(\displaystyle\int_0^1 \frac {1}{1+x^2}dx\).
Chapter 13#
Integration by substitution#
Find \(\displaystyle I = \int xe^{x^2}dx.\)
Find the following integrals:
(i) \(\displaystyle \int x(x^2-3)^5 dx\);
(ii) \(\displaystyle \int (3x-1)^5 dx\);
(iii) \(\displaystyle \int \frac{x+1}{(x^2+2x -5)^3} dx\);
(iv) \(\displaystyle \int \frac{2x}{(2x^2-7)^2} dx\);
(v) \(\displaystyle \int 2x\sqrt{3x^2-5} dx\);
(vi) \(\displaystyle \int (x^3+1)^2 dx\);
(vii) \(\displaystyle \int_3^4 \frac{x^2-1}{\sqrt{x^3-3x}} dx\);
(viii) \(\displaystyle \int_0^1 (2x^2 -1)^3 dx\).
Find
(i) \(\displaystyle \int 3\cos(3x) dx\);
(ii) \(\displaystyle \int \sin(2x+3) dx\);
(iii) \(\displaystyle \int \text{ cosec}^3x \cot x dx\);
(iv) \(\displaystyle \int \sec^2(x) \tan^2(x) dx\);
(v) \(\displaystyle \int \sec^5 x\tan(x) dx\);
(vi) \(\displaystyle \int x \text{ cosec}^2(x^2) dx\).
Find \(\displaystyle \int \tan(x) dx\) and \(\displaystyle \int \cot x dx\).
Find \(\displaystyle \int \cos(2x+3) dx\) and more generally \(\displaystyle \int \cos(ax+b)dx\) for \(a \neq 0\).
Definite integration by substitution#
Evaluate the following definite integrals:
(i) \(\displaystyle \int_0^{\frac{1}{2}} \frac{x}{\sqrt{1-x^2}}dx\);
(ii) \(\displaystyle \int_{-1}^{0} x(x^2-1)^4 dx\);
(iii) \(\displaystyle \int_{-1}^0 x(x+2)^2dx\);
(iv) \(\displaystyle \int_1^2 \frac{x}{(x^2+1)^2} dx\);
(v) \(\displaystyle \int_2^3 \frac{x-1}{(2x^2 -4x +1)^\frac 32} dx\);
(vi) \(\displaystyle \int_0^1 (2x-3)(x^2-3x+7)^2 dx\);
(vii) \(\displaystyle \int_{-\frac \pi2}^0 \cos(x) \sin(x) dx\);
(viii) \(\displaystyle \int_0^{\frac \pi 3} \sin(3x) \cos^2(3x) dx\);
(ix) \(\displaystyle \int_0^{\frac \pi 2} \cos(x) \sqrt{\sin(x)} dx\);
(x) \(\displaystyle \int_\pi^{2\pi} \frac{\cos\sqrt{x}}{\sqrt{x}} dx\).
Calculate the area enclosed by the curve \(\displaystyle y = \frac{x}{\sqrt{x^2-1}}\), the \(x\)-axis, \(x = 2\) and \(x = 3\).
Find the following integrals:
(i) \(\displaystyle \int \frac{1}{4x} dx\);
(ii) \(\displaystyle \int \frac 1{2x+8} dx\);
(iIi) \(\displaystyle \int \frac{2x+1}{x^2 +x -2} dx\);
(iv) \(\displaystyle \int \frac{2x-3}{3x^2 -9x +4} dx\);
(v) \(\displaystyle \int \frac{x}{x+2} dx\);
(vi) \(\displaystyle \int \frac{3x}{2x+3} dx\);
(vii) \(\displaystyle \int \frac{2x}{3-x} dx\);
(viii) \(\displaystyle \int \frac{x-1}{2-x} dx\);
(ix) \(\displaystyle \int \cot \frac x2 dx\);
(x) \(\displaystyle \int \cot(2x +1) dx\);
(xi) \(\displaystyle \int -\tan \frac x3 dx\);
(xii) \(\displaystyle \int \frac{1}{x^2+a^2} dx\);
(xiii) \(\displaystyle \int \frac{1}{\sqrt{a^2 -x^2}} dx\);
(xiv) \(\displaystyle \int \frac 1{1+ x^2} dx\);
(xv) \(\displaystyle \int \frac{x}{1+x^2} dx\);
(xvi) \(\displaystyle \int \frac {1+x}{1+x^2} dx\);
(xvii) \(\displaystyle \int \frac{x}{1-x^2} dx\);
(xviii) \(\displaystyle \int \frac {1+x}{\sqrt{1-x^2}} dx\);
(xix) \(\displaystyle \int \frac 1{1- x} dx\ \ (x < 1)\);
(xx) \(\displaystyle \int \frac{1}{1-x} dx\ \ (x > 1)\);
(xxi) \(\displaystyle \int \frac {x}{1+x} dx\).
Evaluate the following definite integrals:
(i) \(\int_e^{e^2} \frac 5x dx\);
(ii) \(\displaystyle \int_3^4 \frac 1{2x-3} dx\);
(iii) \(\displaystyle \int_5^6 \frac{3-2x}{x-4} dx\);
(iv) \(\displaystyle \int_{-\frac \pi4}^{\frac \pi4} \tan(x) dx\);
(v) \(\displaystyle \int_0^{\frac 12} \frac{x}{\sqrt{1-x^2}} dx\);
(vi) \(\displaystyle \int_{-\frac 12}^{\frac 12} \frac {1}{\sqrt{1-x^2}} dx\);
(vii) \(\displaystyle \int_2^3 \frac {1}{(1-x)^2} dx\);
(viii) \(\displaystyle \int_2^3 \frac {x}{(1-x)^2} dx\).
Integration using partial fractions#
Find
(i) \(\displaystyle \int \frac{1}{x(x-2)} dx\);
(ii) \(\displaystyle \int \frac {1}{(x+3)(5x-2)} dx\);
(iii) \(\displaystyle \int \frac {7x+2}{3x^3 +x^2} dx\);
(iv) \(\displaystyle \int \frac x{16-x^2} dx\);
(v) \(\displaystyle \int \frac{1}{x^2 -4x -5} dx\);
(vi) \(\displaystyle \int \frac{x-2}{x^2 -4x -5} dx\);
(vii) \(\displaystyle \int \frac{2x^2 +2x +3}{(x+2)(x^2+3)} dx\);
(viii) \(\displaystyle \int \frac{22 -16 x}{(3+x)(2-x)(4-x)} dx\).
Find \(\displaystyle \int \frac{x^6 +x^5 -7x^4 -13x^3 +2x^2 -13x-33}{((x+2)^2+1)(x-3)} dx\).
Evaluate
(i) \(\displaystyle \int_0^2 \frac{4x-33}{(2x+1)(x^2-9)} dx\);
(ii) \(\displaystyle \int_0^1 \frac{5x +2}{(x-2)^2(x+1)} dx\).
Integration by parts#
Find
(i) \(\displaystyle \int x \cos(x)dx\);
(ii) \(\displaystyle \int 2x \sin(x) dx\);
(iii) \(\displaystyle\int x\sin(2x) dx\);
(iv) \(\displaystyle \int x \cos(2x+1) dx\);
(v) \(\displaystyle \int_0^\pi x^2 \cos(x) dx\);
(vi) \(\displaystyle \int_0^\frac \pi 2 x \sin(x)dx\).
Find
(i) \(\displaystyle \int_0^1 xe^{-x} dx\);
(ii) \(\displaystyle \int xe^{2x}dx\);
(iii) \(\displaystyle \int_0^{-\ln 4} x^2 e^{4x-1} dx \);
(iv) \(\displaystyle \int x^2 e^{-x} dx \);
(v) \(\displaystyle \int x^3e^x dx \);
(vi) \(\displaystyle \int_0^1 t^2e^t dt \);
(vii) \(\displaystyle \int_0^2 \theta e^{2\theta} d\theta \);
(viii) \(\displaystyle \int t^3 e^{-t^2} dt\).
Find
(i) \(\displaystyle \int x^2 \ln(x) dx\);
(ii) \(\displaystyle \int \ln(2x) dx\);
(iii) \(\displaystyle \int_1^9 \sqrt{y} \ln(y)dy \);
(iv) \(\displaystyle \int_1^2 t^2 \ln(t) dt\).
Find \(\displaystyle \int \tan^{-1} x dx\) and\(\displaystyle \int x \tan^{-1} x dx\).
Find
(i) \(\displaystyle \int e^x \cos\left(\frac 12x\right) dx\);
(ii) \(\displaystyle \int e^x \cos(x) dx \);
(iii) \(\displaystyle \int e^{-x} \sin(x) dx \);
(iv) \(\displaystyle \int_0^{\frac \pi 2} e^{2x}\cos(x) dx\).
Trig powers integration#
Find
(i) \(\displaystyle \int \cos^3 (x) dx\);
(ii) \(\displaystyle \int \sin^4(t)dt\);
(iii) \(\displaystyle\int \cos^4(x) dx \);
(iv) \(\displaystyle \int \sin^5 (x)dx\).
Find \(\displaystyle \int x(x+1)^7 dx\).
Find \(\displaystyle \int x^2 \sqrt{1-x^2}dx\) by using the substitution \(x = \sin(\theta)\).