Chapter 7

Integrals

Every example and exercise of the chapter, worked in full. Each page gives the question, the complete solution, the reason behind every step, and the basics it rests on.

Interactive lesson All chapters Past paper questions (46)

Examples

Example 1Anti derivatives of $\cos 2x$ ⁠, $3x^2 + 4x^3$ and $\frac{1}{x}$ by inspectionworked
Example 2 $\int \frac{x^3 - 1}{x^2}\,dx$ ⁠, $\int (x^{2/3} + 1)\,dx$ ⁠, $\int (x^{3/2} + 2e^x - \frac1x)\,dx$ worked
Example 3 $\int(\sin x + \cos x)\,dx$ ⁠, $\int \operatorname{cosec} x(\operatorname{cosec} x + \cot x)\,dx$ ⁠, $\int\frac{1 - \sin x}{\cos^2 x}\,dx$ worked
Example 4The anti derivative of $4x^3 - 6$ with $F(0) = 3$ worked
Example 5Substitution: $\sin mx$ ⁠, $2x\sin(x^2+1)$ ⁠, $\frac{\tan^4\sqrt x\,\sec^2\sqrt x}{\sqrt x}$ ⁠, $\frac{\sin(\tan^{-1}x)}{1+x^2}$ worked
Example 6 $\int\sin^3x\cos^2x\,dx$ ⁠, $\int\frac{\sin x}{\sin(x+a)}\,dx$ ⁠, $\int\frac{dx}{1+\tan x}$ worked
Example 7 $\int\cos^2x\,dx$ ⁠, $\int\sin 2x\cos 3x\,dx$ ⁠, $\int\sin^3x\,dx$ worked
Example 8 $\int\frac{dx}{x^2 - 16}$ and $\int\frac{dx}{\sqrt{2x - x^2}}$ worked
Example 9 $\int\frac{dx}{x^2 - 6x + 13}$ ⁠, $\int\frac{dx}{3x^2 + 13x - 10}$ ⁠, $\int\frac{dx}{\sqrt{5x^2 - 2x}}$ worked
Example 10 $\int\frac{x + 2}{2x^2 + 6x + 5}\,dx$ and $\int\frac{x + 3}{\sqrt{5 - 4x - x^2}}\,dx$ worked
Example 11 $\int\frac{dx}{(x + 1)(x + 2)}$ by partial fractionsworked
Example 12 $\int\frac{x^2+1}{x^2-5x+6}\,dx$ worked
Example 13 $\int\frac{3x-2}{(x+1)^2(x+3)}\,dx$ worked
Example 14 $\int\frac{x^2}{(x^2+1)(x^2+4)}\,dx$ worked
Example 15 $\int\frac{(3\sin\phi-2)\cos\phi}{5-\cos^2\phi-4\sin\phi}\,d\phi$ worked
Example 16 $\int\frac{x^2+x+1}{(x+2)(x^2+1)}\,dx$ worked
Example 17 $\int x\cos x\,dx$ worked
Example 18 $\int\log x\,dx$ worked
Example 19 $\int x\,e^x\,dx$ worked
Example 20 $\int\frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx$ worked
Example 21 $\int e^x\sin x\,dx$ worked
Example 22 $\int e^x\big[f(x)+f'(x)\big]dx$ ⁠: two integralsworked
Example 23 $\int\sqrt{x^2+2x+5}\,dx$ worked
Example 24 $\int\sqrt{3-2x-x^2}\,dx$ worked
Example 25Four definite integrals by the second fundamental theoremworked
Example 26 $\int_{-1}^{1}5x^4\sqrt{x^5+1}\,dx$ worked
Example 27 $\int_0^1\frac{\tan^{-1}x}{1+x^2}\,dx$ worked
Example 28 $\int_{-1}^{2}\left|x^3-x\right|dx$ worked
Example 29 $\int_{-\pi/4}^{\pi/4}\sin^2x\,dx$ worked
Example 30 $\int_0^{\pi}\frac{x\sin x}{1+\cos^2x}\,dx$ worked
Example 31 $\int_{-1}^{1}\sin^5x\cos^4x\,dx$ worked
Example 32 $\int_0^{\pi/2}\frac{\sin^4x}{\sin^4x+\cos^4x}\,dx$ worked
Example 33 $\int_{\pi/6}^{\pi/3}\frac{dx}{1+\sqrt{\tan x}}$ worked
Example 34 $\int_0^{\pi/2}\log\sin x\,dx$ worked
Example 35 $\int\cos 6x\sqrt{1+\sin 6x}\,dx$ worked
Example 36 $\int\frac{(x^4-x)^{1/4}}{x^5}\,dx$ worked
Example 37 $\int\frac{x^4}{(x-1)(x^2+1)}\,dx$ worked
Example 38 $\int\left[\log(\log x)+\frac{1}{(\log x)^2}\right]dx$ worked
Example 39 $\int\left(\sqrt{\cot x}+\sqrt{\tan x}\right)dx$ worked
Example 40 $\int\frac{\sin 2x\cos 2x}{\sqrt{9-\cos^4 2x}}\,dx$ worked
Example 41 $\int_{-1}^{3/2}|x\sin\pi x|\,dx$ worked
Example 42 $\int_0^{\pi}\frac{x\,dx}{a^2\cos^2x+b^2\sin^2x}$ worked

Exercise 7.1

Exercise 7.1 · Q1An anti derivative of $\sin 2x$ worked
Exercise 7.1 · Q2An anti derivative of $\cos 3x$ worked
Exercise 7.1 · Q3An anti derivative of $e^{2x}$ worked
Exercise 7.1 · Q4An anti derivative of $(ax+b)^2$ worked
Exercise 7.1 · Q5An anti derivative of $\sin 2x - 4e^{3x}$ worked
Exercise 7.1 · Q6 $\int (4e^{3x} + 1)\,dx$ worked
Exercise 7.1 · Q7 $\int x^2\left(1 - \frac{1}{x^2}\right)dx$ worked
Exercise 7.1 · Q8 $\int (ax^2 + bx + c)\,dx$ worked
Exercise 7.1 · Q9 $\int (2x^2 + e^x)\,dx$ worked
Exercise 7.1 · Q10 $\int \left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)^2 dx$ worked
Exercise 7.1 · Q11 $\int \frac{x^3 + 5x^2 - 4}{x^2}\,dx$ worked
Exercise 7.1 · Q12 $\int \frac{x^3+3x+4}{\sqrt{x}}\,dx$ worked
Exercise 7.1 · Q13 $\int \frac{x^3-x^2+x-1}{x-1}\,dx$ worked
Exercise 7.1 · Q14 $\int (1-x)\sqrt{x}\,dx$ worked
Exercise 7.1 · Q15 $\int \sqrt{x}\,(3x^2+2x+3)\,dx$ worked
Exercise 7.1 · Q16 $\int (2x - 3\cos x + e^x)\,dx$ worked
Exercise 7.1 · Q17 $\int (2x^2 - 3\sin x + 5\sqrt{x})\,dx$ worked
Exercise 7.1 · Q18 $\int \sec x\,(\sec x + \tan x)\,dx$ worked
Exercise 7.1 · Q19 $\int \frac{\sec^2 x}{\operatorname{cosec}^2 x}\,dx$ worked
Exercise 7.1 · Q20 $\int \frac{2 - 3\sin x}{\cos^2 x}\,dx$ worked
Exercise 7.1 · Q21 (MCQ)Antiderivative of $\sqrt{x} + \frac{1}{\sqrt{x}}$ worked
Exercise 7.1 · Q22 (MCQ) $f'(x) = 4x^3 - \frac{3}{x^4}$ ⁠, $f(2) = 0$ worked

Exercise 7.2

Exercise 7.2 · Q1 $\int \frac{2x}{1+x^2}\,dx$ worked
Exercise 7.2 · Q2 $\int \frac{(\log x)^2}{x}\,dx$ worked
Exercise 7.2 · Q3 $\int \frac{dx}{x+x\log x}$ worked
Exercise 7.2 · Q4 $\int \sin x\,\sin(\cos x)\,dx$ worked
Exercise 7.2 · Q5 $\int \sin(ax+b)\cos(ax+b)\,dx$ worked
Exercise 7.2 · Q6 $\int \sqrt{ax+b}\,dx$ worked
Exercise 7.2 · Q7 $\int x\sqrt{x+2}\,dx$ worked
Exercise 7.2 · Q8 $\int x\sqrt{1+2x^2}\,dx$ worked
Exercise 7.2 · Q9 $\int (4x+2)\sqrt{x^2+x+1}\,dx$ worked
Exercise 7.2 · Q10 $\int \frac{dx}{x-\sqrt{x}}$ worked
Exercise 7.2 · Q11 $\int \frac{x}{\sqrt{x+4}}\,dx$ worked
Exercise 7.2 · Q12 $\int (x^3-1)^{1/3}x^5\,dx$ worked
Exercise 7.2 · Q13 $\int \frac{x^2}{(2+3x^3)^3}\,dx$ worked
Exercise 7.2 · Q14 $\int\frac{dx}{x(\log x)^m}$ ⁠, $m \ne 1$ worked
Exercise 7.2 · Q15 $\int\frac{x}{9-4x^2}\,dx$ worked
Exercise 7.2 · Q16 $\int e^{2x+3}\,dx$ worked
Exercise 7.2 · Q17 $\int\frac{x}{e^{x^2}}\,dx$ worked
Exercise 7.2 · Q18 $\int\frac{e^{\tan^{-1}x}}{1+x^2}\,dx$ worked
Exercise 7.2 · Q19 $\int\frac{e^{2x}-1}{e^{2x}+1}\,dx$ worked
Exercise 7.2 · Q20 $\int\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}\,dx$ worked
Exercise 7.2 · Q21 $\int\tan^2(2x-3)\,dx$ worked
Exercise 7.2 · Q22 $\int\sec^2(7-4x)\,dx$ worked
Exercise 7.2 · Q23 $\int\frac{\sin^{-1}x}{\sqrt{1-x^2}}\,dx$ worked
Exercise 7.2 · Q24 $\int\frac{2\cos x-3\sin x}{6\cos x+4\sin x}\,dx$ worked
Exercise 7.2 · Q25 $\int\frac{dx}{\cos^2x\,(1-\tan x)^2}$ worked
Exercise 7.2 · Q26 $\int\frac{\cos\sqrt{x}}{\sqrt{x}}\,dx$ worked
Exercise 7.2 · Q27 $\int\sqrt{\sin 2x}\,\cos 2x\,dx$ worked
Exercise 7.2 · Q28 $\int\frac{\cos x}{\sqrt{1+\sin x}}\,dx$ worked
Exercise 7.2 · Q29 $\int\cot x\,\log\sin x\,dx$ worked
Exercise 7.2 · Q30 $\int\frac{\sin x}{1+\cos x}\,dx$ worked
Exercise 7.2 · Q31 $\int\frac{\sin x}{(1+\cos x)^2}\,dx$ worked
Exercise 7.2 · Q32 $\int\frac{1}{1+\cot x}\,dx$ worked
Exercise 7.2 · Q33 $\int\frac{1}{1-\tan x}\,dx$ worked
Exercise 7.2 · Q34 $\int\frac{\sqrt{\tan x}}{\sin x\cos x}\,dx$ worked
Exercise 7.2 · Q35 $\int\frac{(1+\log x)^2}{x}\,dx$ worked
Exercise 7.2 · Q36 $\int\frac{(x+1)(x+\log x)^2}{x}\,dx$ worked
Exercise 7.2 · Q37 $\int\frac{x^3\sin(\tan^{-1}x^4)}{1+x^8}\,dx$ worked
Exercise 7.2 · Q38 (MCQ) $\int\frac{10x^9+10^x\log_e 10}{x^{10}+10^x}\,dx$ worked
Exercise 7.2 · Q39 (MCQ) $\int\frac{dx}{\sin^2x\cos^2x}$ worked

Exercise 7.3

Exercise 7.3 · Q1 $\int\sin^2(2x+5)\,dx$ worked
Exercise 7.3 · Q2 $\int\sin 3x\cos 4x\,dx$ worked
Exercise 7.3 · Q3 $\int\cos 2x\cos 4x\cos 6x\,dx$ worked
Exercise 7.3 · Q4 $\int\sin^3(2x+1)\,dx$ worked
Exercise 7.3 · Q5 $\int\sin^3x\cos^3x\,dx$ worked
Exercise 7.3 · Q6 $\int\sin x\sin 2x\sin 3x\,dx$ worked
Exercise 7.3 · Q7 $\int\sin 4x\sin 8x\,dx$ worked
Exercise 7.3 · Q8 $\int\frac{1-\cos x}{1+\cos x}\,dx$ worked
Exercise 7.3 · Q9 $\int\frac{\cos x}{1+\cos x}\,dx$ worked
Exercise 7.3 · Q10 $\int\sin^4x\,dx$ worked
Exercise 7.3 · Q11 $\int\cos^4 2x\,dx$ worked
Exercise 7.3 · Q12 $\int\frac{\sin^2x}{1+\cos x}\,dx$ worked
Exercise 7.3 · Q13 $\int\frac{\cos 2x-\cos 2\alpha}{\cos x-\cos\alpha}\,dx$ worked
Exercise 7.3 · Q14 $\int\frac{\cos x-\sin x}{1+\sin 2x}\,dx$ worked
Exercise 7.3 · Q15 $\int\tan^3 2x\,\sec 2x\,dx$ worked
Exercise 7.3 · Q16 $\int\tan^4 x\,dx$ worked
Exercise 7.3 · Q17 $\int\frac{\sin^3x+\cos^3x}{\sin^2x\cos^2x}\,dx$ worked
Exercise 7.3 · Q18 $\int\frac{\cos 2x+2\sin^2x}{\cos^2x}\,dx$ worked
Exercise 7.3 · Q19 $\int\frac{dx}{\sin x\cos^3x}$ worked
Exercise 7.3 · Q20 $\int\frac{\cos 2x}{(\cos x+\sin x)^2}\,dx$ worked
Exercise 7.3 · Q21 $\int\sin^{-1}(\cos x)\,dx$ worked
Exercise 7.3 · Q22 $\int\frac{dx}{\cos(x-a)\cos(x-b)}$ worked
Exercise 7.3 · Q23 (MCQ) $\int\frac{\sin^2x-\cos^2x}{\sin^2x\cos^2x}\,dx$ worked
Exercise 7.3 · Q24 (MCQ) $\int\frac{e^x(1+x)}{\cos^2(e^x x)}\,dx$ worked

Exercise 7.4

Exercise 7.4 · Q1 $\int\frac{3x^2}{x^6+1}\,dx$ worked
Exercise 7.4 · Q2 $\int\frac{dx}{\sqrt{1+4x^2}}$ worked
Exercise 7.4 · Q3 $\int\frac{dx}{\sqrt{(2-x)^2+1}}$ worked
Exercise 7.4 · Q4 $\int\frac{dx}{\sqrt{9-25x^2}}$ worked
Exercise 7.4 · Q5 $\int\frac{3x}{1+2x^4}\,dx$ worked
Exercise 7.4 · Q6 $\int\frac{x^2}{1-x^6}\,dx$ worked
Exercise 7.4 · Q7 $\int\frac{x-1}{\sqrt{x^2-1}}\,dx$ worked
Exercise 7.4 · Q8 $\int\frac{x^2}{\sqrt{x^6+a^6}}\,dx$ worked
Exercise 7.4 · Q9 $\int\frac{\sec^2 x}{\sqrt{\tan^2 x+4}}\,dx$ worked
Exercise 7.4 · Q10 $\int\frac{dx}{\sqrt{x^2+2x+2}}$ worked
Exercise 7.4 · Q11 $\int\frac{dx}{9x^2+6x+5}$ worked
Exercise 7.4 · Q12 $\int\frac{dx}{\sqrt{7-6x-x^2}}$ worked
Exercise 7.4 · Q13 $\int\frac{dx}{\sqrt{(x-1)(x-2)}}$ worked
Exercise 7.4 · Q14 $\int\frac{dx}{\sqrt{8 + 3x - x^2}}$ worked
Exercise 7.4 · Q15 $\int\frac{dx}{\sqrt{(x - a)(x - b)}}$ worked
Exercise 7.4 · Q16 $\int\frac{4x + 1}{\sqrt{2x^2 + x - 3}}\,dx$ worked
Exercise 7.4 · Q17 $\int\frac{x + 2}{\sqrt{x^2 - 1}}\,dx$ worked
Exercise 7.4 · Q18 $\int\frac{5x - 2}{1 + 2x + 3x^2}\,dx$ worked
Exercise 7.4 · Q19 $\int\frac{6x + 7}{\sqrt{(x - 5)(x - 4)}}\,dx$ worked
Exercise 7.4 · Q20 $\int\frac{x + 2}{\sqrt{4x - x^2}}\,dx$ worked
Exercise 7.4 · Q21 $\int\frac{x + 2}{\sqrt{x^2 + 2x + 3}}\,dx$ worked
Exercise 7.4 · Q22 $\int\frac{x + 3}{x^2 - 2x - 5}\,dx$ worked
Exercise 7.4 · Q23 $\int\frac{5x + 3}{\sqrt{x^2 + 4x + 10}}\,dx$ worked
Exercise 7.4 · Q24 (MCQ) $\int\frac{dx}{x^2 + 2x + 2}$ worked
Exercise 7.4 · Q25 (MCQ) $\int\frac{dx}{\sqrt{9x - 4x^2}}$ worked

Exercise 7.5

Exercise 7.5 · Q1 $\int\frac{x}{(x+1)(x+2)}\,dx$ worked
Exercise 7.5 · Q2 $\int\frac{1}{x^2-9}\,dx$ worked
Exercise 7.5 · Q3 $\int\frac{3x-1}{(x-1)(x-2)(x-3)}\,dx$ worked
Exercise 7.5 · Q4 $\int\frac{x}{(x-1)(x-2)(x-3)}\,dx$ worked
Exercise 7.5 · Q5 $\int\frac{2x}{x^2+3x+2}\,dx$ worked
Exercise 7.5 · Q6 $\int\frac{1-x^2}{x(1-2x)}\,dx$ worked
Exercise 7.5 · Q7 $\int\frac{x}{(x^2+1)(x-1)}\,dx$ worked
Exercise 7.5 · Q8 $\int\frac{x}{(x-1)^2(x+2)}\,dx$ worked
Exercise 7.5 · Q9 $\int\frac{3x+5}{x^3-x^2-x+1}\,dx$ worked
Exercise 7.5 · Q10 $\int\frac{2x-3}{(x^2-1)(2x+3)}\,dx$ worked
Exercise 7.5 · Q11 $\int\frac{5x}{(x+1)(x^2-4)}\,dx$ worked
Exercise 7.5 · Q12 $\int\frac{x^3+x+1}{x^2-1}\,dx$ worked
Exercise 7.5 · Q13 $\int\frac{2}{(1 - x)(1 + x^2)}\,dx$ worked
Exercise 7.5 · Q14 $\int\frac{3x - 1}{(x + 2)^2}\,dx$ worked
Exercise 7.5 · Q15 $\int\frac{dx}{x^4 - 1}$ worked
Exercise 7.5 · Q16 $\int\frac{dx}{x(x^n + 1)}$ worked
Exercise 7.5 · Q17 $\int\frac{\cos x\,dx}{(1 - \sin x)(2 - \sin x)}$ worked
Exercise 7.5 · Q18 $\int\frac{(x^2 + 1)(x^2 + 2)}{(x^2 + 3)(x^2 + 4)}\,dx$ worked
Exercise 7.5 · Q19 $\int\frac{2x}{(x^2 + 1)(x^2 + 3)}\,dx$ worked
Exercise 7.5 · Q20 $\int\frac{dx}{x(x^4 - 1)}$ worked
Exercise 7.5 · Q21 $\int\frac{dx}{e^x - 1}$ worked
Exercise 7.5 · Q22 (MCQ) $\int\frac{x\,dx}{(x - 1)(x - 2)}$ worked
Exercise 7.5 · Q23 (MCQ) $\int\frac{dx}{x(x^2 + 1)}$ worked

Exercise 7.6

Exercise 7.6 · Q1 $\int x\sin x\,dx$ worked
Exercise 7.6 · Q2 $\int x\sin 3x\,dx$ worked
Exercise 7.6 · Q3 $\int x^2e^x\,dx$ worked
Exercise 7.6 · Q4 $\int x\log x\,dx$ worked
Exercise 7.6 · Q5 $\int x\log 2x\,dx$ worked
Exercise 7.6 · Q6 $\int x^2\log x\,dx$ worked
Exercise 7.6 · Q7 $\int x\sin^{-1}x\,dx$ worked
Exercise 7.6 · Q8 $\int x\tan^{-1}x\,dx$ worked
Exercise 7.6 · Q9 $\int x\cos^{-1}x\,dx$ worked
Exercise 7.6 · Q10 $\int(\sin^{-1}x)^2\,dx$ worked
Exercise 7.6 · Q11 $\int\frac{x\cos^{-1}x}{\sqrt{1-x^2}}\,dx$ worked
Exercise 7.6 · Q12 $\int x\sec^2x\,dx$ worked
Exercise 7.6 · Q13 $\int\tan^{-1}x\,dx$ worked
Exercise 7.6 · Q14 $\int x(\log x)^2\,dx$ worked
Exercise 7.6 · Q15 $\int(x^2+1)\log x\,dx$ worked
Exercise 7.6 · Q16 $\int e^x(\sin x+\cos x)\,dx$ worked
Exercise 7.6 · Q17 $\int\frac{xe^x}{(1+x)^2}\,dx$ worked
Exercise 7.6 · Q18 $\int e^x\frac{1+\sin x}{1+\cos x}\,dx$ worked
Exercise 7.6 · Q19 $\int e^x\left(\frac1x-\frac1{x^2}\right)dx$ worked
Exercise 7.6 · Q20 $\int\frac{(x-3)e^x}{(x-1)^3}\,dx$ worked
Exercise 7.6 · Q21 $\int e^{2x}\sin x\,dx$ worked
Exercise 7.6 · Q22 $\int\sin^{-1}\frac{2x}{1+x^2}\,dx$ worked
Exercise 7.6 · Q23 (MCQ) $\int x^2e^{x^3}\,dx$ worked
Exercise 7.6 · Q24 (MCQ) $\int e^x\sec x(1+\tan x)\,dx$ worked

Exercise 7.7

Exercise 7.7 · Q1 $\int\sqrt{4 - x^2}\,dx$ worked
Exercise 7.7 · Q2 $\int\sqrt{1 - 4x^2}\,dx$ worked
Exercise 7.7 · Q3 $\int\sqrt{x^2 + 4x + 6}\,dx$ worked
Exercise 7.7 · Q4 $\int\sqrt{x^2 + 4x + 1}\,dx$ worked
Exercise 7.7 · Q5 $\int\sqrt{1 - 4x - x^2}\,dx$ worked
Exercise 7.7 · Q6 $\int\sqrt{x^2 + 4x - 5}\,dx$ worked
Exercise 7.7 · Q7 $\int\sqrt{1 + 3x - x^2}\,dx$ worked
Exercise 7.7 · Q8 $\int\sqrt{x^2 + 3x}\,dx$ worked
Exercise 7.7 · Q9 $\int\sqrt{1 + \frac{x^2}{9}}\,dx$ worked
Exercise 7.7 · Q10 (MCQ) $\int\sqrt{1 + x^2}\,dx$ worked
Exercise 7.7 · Q11 (MCQ) $\int\sqrt{x^2 - 8x + 7}\,dx$ worked

Exercise 7.8

Exercise 7.8 · Q1 $\int_{-1}^{1}(x+1)\,dx$ worked
Exercise 7.8 · Q2 $\int_2^3\frac{1}{x}\,dx$ worked
Exercise 7.8 · Q3 $\int_1^2(4x^3 - 5x^2 + 6x + 9)\,dx$ worked
Exercise 7.8 · Q4 $\int_0^{\pi/4}\sin 2x\,dx$ worked
Exercise 7.8 · Q5 $\int_0^{\pi/2}\cos 2x\,dx$ worked
Exercise 7.8 · Q6 $\int_4^5 e^x\,dx$ worked
Exercise 7.8 · Q7 $\int_0^{\pi/4}\tan x\,dx$ worked
Exercise 7.8 · Q8 $\int_{\pi/6}^{\pi/4}\operatorname{cosec}x\,dx$ worked
Exercise 7.8 · Q9 $\int_0^1\frac{dx}{\sqrt{1 - x^2}}$ worked
Exercise 7.8 · Q10 $\int_0^1\frac{dx}{1 + x^2}$ worked
Exercise 7.8 · Q11 $\int_2^3\frac{dx}{x^2 - 1}$ worked
Exercise 7.8 · Q12 $\int_0^{\pi/2}\cos^2 x\,dx$ worked
Exercise 7.8 · Q13 $\int_2^3 \frac{x\,dx}{x^2 + 1}$ worked
Exercise 7.8 · Q14 $\int_0^1 \frac{2x + 3}{5x^2 + 1}\,dx$ worked
Exercise 7.8 · Q15 $\int_0^1 x\,e^{x^2}\,dx$ worked
Exercise 7.8 · Q16 $\int_1^2 \frac{5x^2}{x^2 + 4x + 3}\,dx$ worked
Exercise 7.8 · Q17 $\int_0^{\pi/4}(2\sec^2 x + x^3 + 2)\,dx$ worked
Exercise 7.8 · Q18 $\int_0^{\pi}\left(\sin^2\frac{x}{2} - \cos^2\frac{x}{2}\right)dx$ worked
Exercise 7.8 · Q19 $\int_0^2 \frac{6x + 3}{x^2 + 4}\,dx$ worked
Exercise 7.8 · Q20 $\int_0^1\left(x e^x + \sin\frac{\pi x}{4}\right)dx$ worked
Exercise 7.8 · Q21 (MCQ) $\int_1^{\sqrt3}\frac{dx}{1 + x^2}$ worked
Exercise 7.8 · Q22 (MCQ) $\int_0^{2/3}\frac{dx}{4 + 9x^2}$ worked

Exercise 7.9

Exercise 7.9 · Q1 $\int_0^1\frac{x}{x^2+1}\,dx$ worked
Exercise 7.9 · Q2 $\int_0^{\pi/2}\sqrt{\sin\phi}\,\cos^5\phi\,d\phi$ worked
Exercise 7.9 · Q3 $\int_0^1\sin^{-1}\frac{2x}{1+x^2}\,dx$ worked
Exercise 7.9 · Q4 $\int_0^2 x\sqrt{x+2}\,dx$ worked
Exercise 7.9 · Q5 $\int_0^{\pi/2}\frac{\sin x}{1+\cos^2x}\,dx$ worked
Exercise 7.9 · Q6 $\int_0^2\frac{dx}{x+4-x^2}$ worked
Exercise 7.9 · Q7 $\int_{-1}^1\frac{dx}{x^2+2x+5}$ worked
Exercise 7.9 · Q8 $\int_1^2\left(\frac{1}{x}-\frac{1}{2x^2}\right)e^{2x}dx$ worked
Exercise 7.9 · Q9 (MCQ) $\int_{1/3}^1\frac{(x-x^3)^{1/3}}{x^4}\,dx$ worked
Exercise 7.9 · Q10 (MCQ) $f(x)=\int_0^x t\sin t\,dt$ ⁠, find $f'(x)$ worked

Exercise 7.10

Exercise 7.10 · Q1 $\int_0^{\pi/2}\cos^2x\,dx$ worked
Exercise 7.10 · Q2 $\int_0^{\pi/2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx$ worked
Exercise 7.10 · Q3 $\int_0^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$ worked
Exercise 7.10 · Q4 $\int_0^{\pi/2}\frac{\cos^5x}{\sin^5x+\cos^5x}\,dx$ worked
Exercise 7.10 · Q5 $\int_{-5}^{5}|x+2|\,dx$ worked
Exercise 7.10 · Q6 $\int_2^8|x-5|\,dx$ worked
Exercise 7.10 · Q7 $\int_0^1x(1-x)^n\,dx$ worked
Exercise 7.10 · Q8 $\int_0^{\pi/4}\log(1+\tan x)\,dx$ worked
Exercise 7.10 · Q9 $\int_0^2x\sqrt{2-x}\,dx$ worked
Exercise 7.10 · Q10 $\int_0^{\pi/2}(2\log\sin x-\log\sin 2x)\,dx$ worked
Exercise 7.10 · Q11 $\int_{-\pi/2}^{\pi/2}\sin^2x\,dx$ worked
Exercise 7.10 · Q12 $\int_0^{\pi}\frac{x}{1+\sin x}\,dx$ worked
Exercise 7.10 · Q13 $\int_{-\pi/2}^{\pi/2}\sin^7x\,dx$ worked
Exercise 7.10 · Q14 $\int_0^{2\pi}\cos^5x\,dx$ worked
Exercise 7.10 · Q15 $\int_0^{\pi/2}\frac{\sin x-\cos x}{1+\sin x\cos x}\,dx$ worked
Exercise 7.10 · Q16 $\int_0^{\pi}\log(1+\cos x)\,dx$ worked
Exercise 7.10 · Q17 $\int_0^a\frac{\sqrt x}{\sqrt x+\sqrt{a-x}}\,dx$ worked
Exercise 7.10 · Q18 $\int_0^4|x-1|\,dx$ worked
Exercise 7.10 · Q19 $\int_0^a fg\,dx=2\int_0^a f\,dx$ worked
Exercise 7.10 · Q20 (MCQ) $\int_{-\pi/2}^{\pi/2}(x^3+x\cos x+\tan^5x+1)\,dx$ worked
Exercise 7.10 · Q21 (MCQ) $\int_0^{\pi/2}\log\frac{4+3\sin x}{4+3\cos x}\,dx$ worked

Miscellaneous Exercise

Miscellaneous · Q1 $\int\frac{dx}{x-x^3}$ worked
Miscellaneous · Q2 $\int\frac{dx}{\sqrt{x+a}+\sqrt{x+b}}$ worked
Miscellaneous · Q3 $\int\frac{dx}{x\sqrt{ax-x^2}}$ worked
Miscellaneous · Q4 $\int\frac{dx}{x^2(x^4+1)^{3/4}}$ worked
Miscellaneous · Q5 $\int\frac{dx}{x^{1/2}+x^{1/3}}$ worked
Miscellaneous · Q6 $\int\frac{5x}{(x+1)(x^2+9)}\,dx$ worked
Miscellaneous · Q7 $\int\frac{\sin x}{\sin(x-a)}\,dx$ worked
Miscellaneous · Q8 $\int\frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}\,dx$ worked
Miscellaneous · Q9 $\int\frac{\cos x}{\sqrt{4-\sin^2x}}\,dx$ worked
Miscellaneous · Q10 $\int\frac{\sin^8x-\cos^8x}{1-2\sin^2x\cos^2x}\,dx$ worked
Miscellaneous · Q11 $\int\frac{dx}{\cos(x+a)\cos(x+b)}$ worked
Miscellaneous · Q12 $\int\frac{x^3}{\sqrt{1-x^8}}\,dx$ worked
Miscellaneous · Q13 $\int\frac{e^x}{(1+e^x)(2+e^x)}\,dx$ worked
Miscellaneous · Q14 $\int\frac{dx}{(x^2+1)(x^2+4)}$ worked
Miscellaneous · Q15 $\int\cos^3x\,e^{\log\sin x}\,dx$ worked
Miscellaneous · Q16 $\int e^{3\log x}(x^4+1)^{-1}\,dx$ worked
Miscellaneous · Q17 $\int f'(ax+b)\,[f(ax+b)]^n\,dx$ worked
Miscellaneous · Q18 $\int\frac{dx}{\sqrt{\sin^3x\,\sin(x+\alpha)}}$ worked
Miscellaneous · Q19 $\int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}\,dx$ worked
Miscellaneous · Q20 $\int\frac{2+\sin 2x}{1+\cos 2x}\,e^x\,dx$ worked
Miscellaneous · Q21 $\int\frac{x^2+x+1}{(x+1)^2(x+2)}\,dx$ worked
Miscellaneous · Q22 $\int\tan^{-1}\sqrt{\frac{1-x}{1+x}}\,dx$ worked
Miscellaneous · Q23 $\int\frac{\sqrt{x^2+1}\,[\log(x^2+1)-2\log x]}{x^4}\,dx$ worked
Miscellaneous · Q24 $\int_{\pi/2}^{\pi}e^x\left(\frac{1-\sin x}{1-\cos x}\right)dx$ worked
Miscellaneous · Q25 $\int_0^{\pi/4}\frac{\sin x\cos x}{\cos^4x+\sin^4x}\,dx$ worked
Miscellaneous · Q26 $\int_0^{\pi/2}\frac{\cos^2x\,dx}{\cos^2x+4\sin^2x}$ worked
Miscellaneous · Q27 $\int_{\pi/6}^{\pi/3}\frac{\sin x+\cos x}{\sqrt{\sin 2x}}\,dx$ worked
Miscellaneous · Q28 $\int_0^1\frac{dx}{\sqrt{1+x}-\sqrt{x}}$ worked
Miscellaneous · Q29 $\int_0^{\pi/4}\frac{\sin x+\cos x}{9+16\sin 2x}\,dx$ worked
Miscellaneous · Q30 $\int_0^{\pi/2}\sin 2x\,\tan^{-1}(\sin x)\,dx$ worked
Miscellaneous · Q31 $\int_1^4\big(|x-1|+|x-2|+|x-3|\big)\,dx$ worked
Miscellaneous · Q32 $\int_1^3\frac{dx}{x^2(x+1)}=\frac23+\log\frac23$ worked
Miscellaneous · Q33 $\int_0^1 x\,e^x\,dx=1$ worked
Miscellaneous · Q34 $\int_{-1}^{1}x^{17}\cos^4x\,dx=0$ worked
Miscellaneous · Q35 $\int_0^{\pi/2}\sin^3x\,dx=\frac23$ worked
Miscellaneous · Q36 $\int_0^{\pi/4}2\tan^3x\,dx=1-\log 2$ worked
Miscellaneous · Q37 $\int_0^1\sin^{-1}x\,dx=\frac{\pi}{2}-1$ worked
Miscellaneous · Q38 (MCQ) $\int\frac{dx}{e^x+e^{-x}}$ worked
Miscellaneous · Q39 (MCQ) $\int\frac{\cos 2x}{(\sin x+\cos x)^2}\,dx$ worked
Miscellaneous · Q40 (MCQ) $\int_a^b x f(x)\,dx$ when $f(a+b-x)=f(x)$ worked

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