Exercise 7.2 Practice
Integration by Substitution – NCERT Solutions
Q1
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Integrate the function: $\frac{2x}{x^2+4}$
Let $x^2+4 = t$.
Differentiating w.r.t $x$: $2x dx = dt$.
Integral becomes $\int \frac{1}{t} dt = \log|t| + C$.
Substitute back: $\log|x^2+4| + C$.
$\boxed{\log(x^2+4) + C}$
Q2
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Integrate the function: $\frac{(\log x)^3}{x}$
Let $\log x = t \Rightarrow \frac{1}{x} dx = dt$.
Integral becomes $\int t^3 dt = \frac{t^4}{4} + C$.
$\boxed{\frac{(\log x)^4}{4} + C}$
Q3
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Integrate the function: $\frac{1}{x(2 + 3\log x)}$
Let $2+3\log x = t \Rightarrow \frac{3}{x} dx = dt \Rightarrow \frac{1}{x} dx = \frac{dt}{3}$.
Integral becomes $\frac{1}{3} \int \frac{1}{t} dt = \frac{1}{3}\log|t| + C$.
$\boxed{\frac{1}{3}\log|2+3\log x| + C}$
Q4
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Integrate the function: $\cos x \cos(\sin x)$
Let $\sin x = t \Rightarrow \cos x dx = dt$.
Integral becomes $\int \cos t dt = \sin t + C$.
$\boxed{\sin(\sin x) + C}$
Q5
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Integrate the function: $\sin(2x+3)\cos(2x+3)$
Let $\sin(2x+3) = t \Rightarrow 2\cos(2x+3) dx = dt$.
Integral: $\frac{1}{2} \int t dt = \frac{1}{2} \frac{t^2}{2} + C$.
$\boxed{\frac{1}{4}\sin^2(2x+3) + C}$
Q6
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Integrate the function: $\sqrt{3x+5}$
$\int (3x+5)^{1/2} dx = \frac{(3x+5)^{3/2}}{3/2 \cdot 3} + C$.
$\boxed{\frac{2}{9}(3x+5)^{3/2} + C}$
Q7
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Integrate the function: $x\sqrt{x+4}$
Let $x+4 = t \Rightarrow x = t-4, dx = dt$.
$\int (t-4)\sqrt{t} dt = \int (t^{3/2} - 4t^{1/2}) dt$.
$\frac{2}{5}t^{5/2} - 4(\frac{2}{3}t^{3/2}) + C$.
$\boxed{\frac{2}{5}(x+4)^{5/2} - \frac{8}{3}(x+4)^{3/2} + C}$
Q8
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Integrate the function: $x\sqrt{1+x^2}$
Let $1+x^2 = t \Rightarrow 2x dx = dt \Rightarrow x dx = dt/2$.
$\frac{1}{2} \int t^{1/2} dt = \frac{1}{2} \frac{2}{3} t^{3/2} + C$.
$\boxed{\frac{1}{3}(1+x^2)^{3/2} + C}$
Q9
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Integrate the function: $(2x+1)\sqrt{x^2+x+5}$
Let $x^2+x+5 = t \Rightarrow (2x+1) dx = dt$.
Integral becomes $\int \sqrt{t} dt$.
$\frac{2}{3} t^{3/2} + C$.
$\boxed{\frac{2}{3}(x^2+x+5)^{3/2} + C}$
Q10
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Integrate the function: $\frac{1}{x + \sqrt{x}}$
Factor out $\sqrt{x}$: $\frac{1}{\sqrt{x}(\sqrt{x}+1)}$.
Let $\sqrt{x}+1 = t \Rightarrow \frac{1}{2\sqrt{x}} dx = dt \Rightarrow \frac{1}{\sqrt{x}} dx = 2dt$.
$2 \int \frac{1}{t} dt = 2\log|t| + C$.
$\boxed{2\log(\sqrt{x}+1) + C}$
Q11
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Integrate the function: $\frac{x}{\sqrt{x+3}}$
Let $x+3 = t \Rightarrow x = t-3, dx = dt$.
$\int \frac{t-3}{\sqrt{t}} dt = \int (t^{1/2} - 3t^{-1/2}) dt$.
$\frac{2}{3}t^{3/2} - 3(2t^{1/2}) + C$.
$\boxed{\frac{2}{3}(x+3)^{3/2} - 6\sqrt{x+3} + C}$
Q12
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Integrate the function: $(x^3+1)^{1/3}x^5$
Let $x^3+1 = t \Rightarrow x^3 = t-1, 3x^2 dx = dt$.
Rewrite $x^5 = x^3 \cdot x^2$. Integral: $\int t^{1/3}(t-1) \frac{dt}{3}$.
$\frac{1}{3} \int (t^{4/3} - t^{1/3}) dt = \frac{1}{3} [\frac{3}{7}t^{7/3} - \frac{3}{4}t^{4/3}] + C$.
$\boxed{\frac{1}{7}(x^3+1)^{7/3} - \frac{1}{4}(x^3+1)^{4/3} + C}$
Q13
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Integrate the function: $\frac{x^2}{(1+x^3)^2}$
Let $1+x^3 = t \Rightarrow 3x^2 dx = dt \Rightarrow x^2 dx = dt/3$.
$\frac{1}{3} \int t^{-2} dt = \frac{1}{3} \frac{t^{-1}}{-1} + C$.
$\boxed{-\frac{1}{3(1+x^3)} + C}$
Q14
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Integrate the function: $\frac{1}{x(\log x)^2}$
Let $\log x = t \Rightarrow \frac{1}{x} dx = dt$.
$\int t^{-2} dt = -t^{-1} + C$.
$\boxed{-\frac{1}{\log x} + C}$
Q15
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Integrate the function: $\frac{x}{1-x^2}$
Let $1-x^2 = t \Rightarrow -2x dx = dt \Rightarrow x dx = -dt/2$.
$-\frac{1}{2} \int \frac{1}{t} dt = -\frac{1}{2} \log|t| + C$.
$\boxed{-\frac{1}{2}\log|1-x^2| + C}$
Q16
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Integrate the function: $e^{5x+2}$
$\int e^{5x+2} dx = \frac{e^{5x+2}}{5} + C$.
$\boxed{\frac{1}{5}e^{5x+2} + C}$
Q17
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Integrate the function: $x e^{x^2}$
Let $x^2 = t \Rightarrow 2x dx = dt \Rightarrow x dx = dt/2$.
$\frac{1}{2} \int e^{t} dt = \frac{1}{2} e^{t} + C$.
$\boxed{\frac{1}{2}e^{x^2} + C}$
Q18
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Integrate the function: $\frac{e^{\sin^{-1}x}}{\sqrt{1-x^2}}$
Let $\sin^{-1}x = t \Rightarrow \frac{1}{\sqrt{1-x^2}} dx = dt$.
$\int e^t dt = e^t + C$.
$\boxed{e^{\sin^{-1}x} + C}$
Q19
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Integrate the function: $\frac{e^{4x}-1}{e^{4x}+1}$
Divide numerator and denominator by $e^{2x}$: $\frac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}$.
Let $e^{2x} + e^{-2x} = t \Rightarrow 2(e^{2x} - e^{-2x}) dx = dt$.
$\frac{1}{2}\int \frac{1}{t} dt = \frac{1}{2}\log|t| + C$.
$\boxed{\frac{1}{2}\log(e^{2x} + e^{-2x}) + C}$
Q20
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Integrate the function: $\frac{e^{3x}-e^{-3x}}{e^{3x}+e^{-3x}}$
Let $e^{3x}+e^{-3x} = t \Rightarrow 3(e^{3x} - e^{-3x}) dx = dt$.
$\frac{1}{3} \int \frac{1}{t} dt = \frac{1}{3} \log|t| + C$.
$\boxed{\frac{1}{3}\log(e^{3x}+e^{-3x}) + C}$
Q21
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Integrate the function: $\tan^2(3x+1)$
Use $\tan^2 \theta = \sec^2 \theta - 1$.
$\int (\sec^2(3x+1) - 1) dx = \frac{\tan(3x+1)}{3} - x + C$.
$\boxed{\frac{1}{3}\tan(3x+1) - x + C}$
Q22
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Integrate the function: $\sec^2(2-5x)$
$\int \sec^2(2-5x) dx = \frac{\tan(2-5x)}{-5} + C$.
$\boxed{-\frac{1}{5}\tan(2-5x) + C}$
Q23
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Integrate the function: $\frac{(\tan^{-1}x)^2}{1+x^2}$
Let $\tan^{-1}x = t \Rightarrow \frac{1}{1+x^2} dx = dt$.
$\int t^2 dt = \frac{t^3}{3} + C$.
$\boxed{\frac{1}{3}(\tan^{-1}x)^3 + C}$
Q24
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Integrate the function: $\frac{\cos x - \sin x}{\cos x + \sin x}$
Let $\cos x + \sin x = t \Rightarrow (-\sin x + \cos x) dx = dt$.
$\int \frac{1}{t} dt = \log|t| + C$.
$\boxed{\log|\cos x + \sin x| + C}$
Q25
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Integrate the function: $\frac{\sec^2 x}{(1+\tan x)^3}$
Let $1+\tan x = t \Rightarrow \sec^2 x dx = dt$.
$\int t^{-3} dt = \frac{t^{-2}}{-2} + C$.
$\boxed{-\frac{1}{2(1+\tan x)^2} + C}$
Q26
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Integrate the function: $\frac{\sin\sqrt{x}}{\sqrt{x}}$
Let $\sqrt{x} = t \Rightarrow \frac{1}{2\sqrt{x}} dx = dt \Rightarrow \frac{1}{\sqrt{x}} dx = 2dt$.
$2 \int \sin t dt = -2\cos t + C$.
$\boxed{-2\cos\sqrt{x} + C}$
Q27
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Integrate the function: $\sqrt{\cos x} \sin x$
Let $\cos x = t \Rightarrow -\sin x dx = dt$.
$-\int t^{1/2} dt = -\frac{2}{3} t^{3/2} + C$.
$\boxed{-\frac{2}{3}(\cos x)^{3/2} + C}$
Q28
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Integrate the function: $\frac{\sin x}{\sqrt{1+\cos x}}$
Let $1+\cos x = t \Rightarrow -\sin x dx = dt$.
$\int -t^{-1/2} dt = -2t^{1/2} + C$.
$\boxed{-2\sqrt{1+\cos x} + C}$
Q29
00:00
Integrate the function: $\tan x \log(\cos x)$
Let $\log(\cos x) = t \Rightarrow \frac{1}{\cos x} (-\sin x) dx = dt \Rightarrow -\tan x dx = dt$.
$\int -t dt = -\frac{t^2}{2} + C$.
$\boxed{-\frac{1}{2}(\log(\cos x))^2 + C}$
Q30
00:00
Integrate the function: $\frac{\cos x}{1+\sin x}$
Let $1+\sin x = t \Rightarrow \cos x dx = dt$.
$\int \frac{1}{t} dt = \log|t| + C$.
$\boxed{\log|1+\sin x| + C}$
Q31
00:00
Integrate the function: $\frac{\cos x}{(1+\sin x)^2}$
Let $1+\sin x = t \Rightarrow \cos x dx = dt$.
$\int t^{-2} dt = \frac{t^{-1}}{-1} + C$.
$\boxed{-\frac{1}{1+\sin x} + C}$
Q32
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Integrate the function: $\frac{1}{1-\cot x}$
$\frac{1}{1-\frac{\cos x}{\sin x}} = \frac{\sin x}{\sin x - \cos x}$.
Multiply/divide by 2: $\frac{1}{2} \frac{2\sin x}{\sin x - \cos x} = \frac{1}{2} \frac{(\sin x - \cos x) + (\sin x + \cos x)}{\sin x - \cos x}$.
$\frac{1}{2} [1 + \frac{\sin x + \cos x}{\sin x - \cos x}]$.
Integrate: $\frac{1}{2}x + \frac{1}{2}\log|\sin x - \cos x| + C$.
$\boxed{\frac{x}{2} + \frac{1}{2}\log|\sin x - \cos x| + C}$
Q33
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Integrate the function: $\frac{1}{1+\tan x}$
$\frac{\cos x}{\cos x + \sin x}$.
$\frac{1}{2} \frac{2\cos x}{\cos x + \sin x} = \frac{1}{2} \frac{(\cos x + \sin x) + (\cos x - \sin x)}{\cos x + \sin x}$.
$\frac{1}{2} [1 + \frac{\cos x - \sin x}{\cos x + \sin x}]$.
$\frac{1}{2}x + \frac{1}{2}\log|\cos x + \sin x| + C$.
$\boxed{\frac{x}{2} + \frac{1}{2}\log|\cos x + \sin x| + C}$
Q34
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Integrate the function: $\frac{\sqrt{\cot x}}{\sin x \cos x}$
Divide num/den by $\sin^2 x$: $\frac{\sqrt{\cot x} \csc^2 x}{\cot x} = \frac{\csc^2 x}{\sqrt{\cot x}}$.
Let $\cot x = t \Rightarrow -\csc^2 x dx = dt$.
$\int -t^{-1/2} dt = -2\sqrt{t} + C$.
$\boxed{-2\sqrt{\cot x} + C}$
Q35
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Integrate the function: $\frac{(2+\log x)^3}{x}$
Let $2+\log x = t \Rightarrow \frac{1}{x} dx = dt$.
$\int t^3 dt = \frac{t^4}{4} + C$.
$\boxed{\frac{(2+\log x)^4}{4} + C}$
Q36
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Integrate the function: $\frac{(x+1)(x+\log x)}{x}$
Rewrite: $(1+\frac{1}{x})(x+\log x)$.
Let $x+\log x = t \Rightarrow (1+\frac{1}{x}) dx = dt$.
$\int t dt = \frac{t^2}{2} + C$.
$\boxed{\frac{(x+\log x)^2}{2} + C}$
Q37
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Integrate the function: $\frac{x^2 \sin(\tan^{-1}x^3)}{1+x^6}$
Let $\tan^{-1}x^3 = t \Rightarrow \frac{1}{1+(x^3)^2} \cdot 3x^2 dx = dt \Rightarrow \frac{x^2}{1+x^6} dx = \frac{dt}{3}$.
$\frac{1}{3} \int \sin t dt = -\frac{1}{3}\cos t + C$.
$\boxed{-\frac{1}{3}\cos(\tan^{-1}x^3) + C}$
Q38
00:00
$\int \frac{5x^4 + 5^x \log_e 5}{x^5 + 5^x} dx$ equals:
(A) $5^x - x^5 + C$
(B) $5^x + x^5 + C$
(C) $(5^x - x^5)^{-1} + C$
(D) $\log(x^5 + 5^x) + C$
(A) $5^x - x^5 + C$
(B) $5^x + x^5 + C$
(C) $(5^x - x^5)^{-1} + C$
(D) $\log(x^5 + 5^x) + C$
Let $x^5 + 5^x = t$.
Derivative: $(5x^4 + 5^x \log_e 5) dx = dt$.
$\int \frac{1}{t} dt = \log|t| + C$.
$\boxed{\text{(D) } \log(x^5 + 5^x) + C}$
Q39
00:00
$\int \frac{\cos 2x}{\sin^2 x \cos^2 x} dx$ equals:
(A) $\tan x + \cot x + C$
(B) $-\cot x - \tan x + C$
(C) $\tan x - \cot x + C$
(D) $\cot x - \tan x + C$
(A) $\tan x + \cot x + C$
(B) $-\cot x - \tan x + C$
(C) $\tan x - \cot x + C$
(D) $\cot x - \tan x + C$
$\cos 2x = \cos^2 x - \sin^2 x$.
$\int \frac{\cos^2 x - \sin^2 x}{\sin^2 x \cos^2 x} dx = \int (\frac{1}{\sin^2 x} - \frac{1}{\cos^2 x}) dx$.
$\int (\csc^2 x - \sec^2 x) dx = -\cot x - \tan x + C$.
$\boxed{\text{(B) } -\cot x - \tan x + C}$