Exercise 7.4 Practice
Integration of some particular functions – NCERT Solutions
Q1
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Integrate the function: $\frac{2x}{x^4+1}$
Let $x^2 = t \Rightarrow 2x dx = dt$.
$\int \frac{dt}{t^2+1} = \tan^{-1} t + C$.
$\boxed{\tan^{-1}(x^2) + C}$
Q2
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Integrate the function: $\frac{1}{\sqrt{9x^2+4}}$
$\int \frac{dx}{\sqrt{(3x)^2+2^2}}$. Let $3x = t \Rightarrow 3dx = dt$.
$\frac{1}{3} \int \frac{dt}{\sqrt{t^2+2^2}} = \frac{1}{3} \log|t+\sqrt{t^2+4}| + C$.
$\boxed{\frac{1}{3}\log|3x+\sqrt{9x^2+4}| + C}$
Q3
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Integrate the function: $\frac{1}{\sqrt{(3-x)^2+4}}$
Let $3-x = t \Rightarrow -dx = dt$.
$-\int \frac{dt}{\sqrt{t^2+2^2}} = -\log|t+\sqrt{t^2+4}| + C$.
$\boxed{-\log|3-x+\sqrt{x^2-6x+13}| + C}$
Q4
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Integrate the function: $\frac{1}{\sqrt{16-9x^2}}$
$\int \frac{dx}{\sqrt{4^2-(3x)^2}}$. Let $3x = t \Rightarrow 3dx = dt$.
$\frac{1}{3} \int \frac{dt}{\sqrt{4^2-t^2}} = \frac{1}{3} \sin^{-1}(\frac{t}{4}) + C$.
$\boxed{\frac{1}{3}\sin^{-1}(\frac{3x}{4}) + C}$
Q5
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Integrate the function: $\frac{x}{1+x^4}$
$\frac{x}{1+(x^2)^2}$. Let $x^2 = t \Rightarrow 2x dx = dt$.
$\frac{1}{2} \int \frac{dt}{1+t^2} = \frac{1}{2} \tan^{-1} t + C$.
$\boxed{\frac{1}{2}\tan^{-1}(x^2) + C}$
Q6
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Integrate the function: $\frac{x^2}{4-x^6}$
Let $x^3 = t \Rightarrow 3x^2 dx = dt$.
$\frac{1}{3} \int \frac{dt}{2^2-t^2} = \frac{1}{3} \cdot \frac{1}{2(2)} \log|\frac{2+t}{2-t}| + C$.
$\boxed{\frac{1}{12}\log|\frac{2+x^3}{2-x^3}| + C}$
Q7
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Integrate the function: $\frac{x+1}{\sqrt{x^2-9}}$
$\int \frac{x}{\sqrt{x^2-9}} dx + \int \frac{1}{\sqrt{x^2-9}} dx$.
First part: Let $x^2-9=t \Rightarrow 2x dx=dt$. $\frac{1}{2}\int t^{-1/2} dt = \sqrt{t}$.
Second part: $\log|x+\sqrt{x^2-9}|$.
$\boxed{\sqrt{x^2-9} + \log|x+\sqrt{x^2-9}| + C}$
Q8
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Integrate the function: $\frac{x^2}{\sqrt{x^6-1}}$
Let $x^3 = t \Rightarrow 3x^2 dx = dt$.
$\frac{1}{3} \int \frac{dt}{\sqrt{t^2-1}} = \frac{1}{3}\log|t+\sqrt{t^2-1}| + C$.
$\boxed{\frac{1}{3}\log|x^3+\sqrt{x^6-1}| + C}$
Q9
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Integrate the function: $\frac{\cos x}{\sqrt{\sin^2 x + 9}}$
Let $\sin x = t \Rightarrow \cos x dx = dt$.
$\int \frac{dt}{\sqrt{t^2+3^2}} = \log|t+\sqrt{t^2+9}| + C$.
$\boxed{\log|\sin x+\sqrt{\sin^2 x+9}| + C}$
Q10
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Integrate the function: $\frac{1}{\sqrt{x^2+4x+5}}$
$\sqrt{x^2+4x+4+1} = \sqrt{(x+2)^2+1}$.
$\int \frac{dx}{\sqrt{(x+2)^2+1}} = \log|x+2+\sqrt{x^2+4x+5}| + C$.
$\boxed{\log|x+2+\sqrt{x^2+4x+5}| + C}$
Q11
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Integrate the function: $\frac{1}{x^2+6x+13}$
$x^2+6x+13 = (x+3)^2 + 4$.
$\int \frac{dx}{(x+3)^2+2^2} = \frac{1}{2} \tan^{-1}(\frac{x+3}{2}) + C$.
$\boxed{\frac{1}{2}\tan^{-1}(\frac{x+3}{2}) + C}$
Q12
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Integrate the function: $\frac{1}{\sqrt{5-4x-x^2}}$
$5-(x^2+4x) = 5-(x^2+4x+4-4) = 9-(x+2)^2$.
$\int \frac{dx}{\sqrt{3^2-(x+2)^2}} = \sin^{-1}(\frac{x+2}{3}) + C$.
$\boxed{\sin^{-1}(\frac{x+2}{3}) + C}$
Q13
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Integrate the function: $\frac{1}{\sqrt{(x-3)(x-4)}}$
$\sqrt{x^2-7x+12} = \sqrt{(x-\frac{7}{2})^2 - \frac{1}{4}}$.
$\log|x-\frac{7}{2} + \sqrt{x^2-7x+12}| + C$.
$\boxed{\log|x-\frac{7}{2} + \sqrt{x^2-7x+12}| + C}$
Q14
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Integrate the function: $\frac{1}{\sqrt{3-2x-x^2}}$
$3-(x^2+2x) = 3-(x^2+2x+1-1) = 4 - (x+1)^2$.
$\sin^{-1}(\frac{x+1}{2}) + C$.
$\boxed{\sin^{-1}(\frac{x+1}{2}) + C}$
Q15
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Integrate the function: $\frac{1}{\sqrt{x^2-8x+7}}$
$\sqrt{x^2-8x+16-9} = \sqrt{(x-4)^2 - 3^2}$.
$\log|x-4 + \sqrt{x^2-8x+7}| + C$.
$\boxed{\log|x-4 + \sqrt{x^2-8x+7}| + C}$
Q16
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Integrate the function: $\frac{2x+3}{\sqrt{x^2+4x+1}}$
$2x+3 = (2x+4) - 1$.
$\int \frac{2x+4}{\sqrt{x^2+4x+1}} dx - \int \frac{1}{\sqrt{(x+2)^2-3}} dx$.
$\boxed{2\sqrt{x^2+4x+1} - \log|x+2+\sqrt{x^2+4x+1}| + C}$
Q17
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Integrate the function: $\frac{x+1}{\sqrt{x^2+1}}$
$\frac{x}{\sqrt{x^2+1}} + \frac{1}{\sqrt{x^2+1}}$.
$\sqrt{x^2+1} + \log|x+\sqrt{x^2+1}| + C$.
$\boxed{\sqrt{x^2+1} + \log|x+\sqrt{x^2+1}| + C}$
Q18
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Integrate the function: $\frac{2x+1}{x^2+2x+5}$
$2x+1 = (2x+2) - 1$.
$\int \frac{2x+2}{x^2+2x+5} dx - \int \frac{dx}{(x+1)^2+2^2}$.
$\boxed{\log|x^2+2x+5| - \frac{1}{2}\tan^{-1}(\frac{x+1}{2}) + C}$
Q19
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Integrate the function: $\frac{x+2}{\sqrt{x^2-2x+3}}$
$x+2 = \frac{1}{2}(2x-2) + 3$.
$\frac{1}{2} \int \frac{2x-2}{\sqrt{x^2-2x+3}} dx + 3 \int \frac{dx}{\sqrt{(x-1)^2+2}}$.
$\boxed{\sqrt{x^2-2x+3} + 3\log|x-1+\sqrt{x^2-2x+3}| + C}$
Q20
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Integrate the function: $\frac{x+3}{\sqrt{5-4x-x^2}}$
$x+3 = -\frac{1}{2}(-2x-4) + 1$.
$-\frac{1}{2}(2\sqrt{5-4x-x^2}) + \int \frac{dx}{\sqrt{3^2-(x+2)^2}}$.
$\boxed{-\sqrt{5-4x-x^2} + \sin^{-1}(\frac{x+2}{3}) + C}$
Q21
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Integrate the function: $\frac{2x+5}{\sqrt{x^2+3x+2}}$
$2x+5 = (2x+3) + 2$.
$2\sqrt{x^2+3x+2} + 2 \int \frac{dx}{\sqrt{(x+3/2)^2-1/4}}$.
$\boxed{2\sqrt{x^2+3x+2} + 2\log|x+\frac{3}{2}+\sqrt{x^2+3x+2}| + C}$
Q22
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Integrate the function: $\frac{x}{x^2+x+1}$
$x = \frac{1}{2}(2x+1) - \frac{1}{2}$.
$\frac{1}{2}\log|x^2+x+1| - \frac{1}{2} \int \frac{dx}{(x+1/2)^2+(\sqrt{3}/2)^2}$.
$\boxed{\frac{1}{2}\log|x^2+x+1| - \frac{1}{\sqrt{3}}\tan^{-1}(\frac{2x+1}{\sqrt{3}}) + C}$
Q23
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Integrate the function: $\frac{3x+1}{\sqrt{2x^2+2x+3}}$
$3x+1 = \frac{3}{4}(4x+2) - \frac{1}{2}$.
$\frac{3}{4} \cdot 2\sqrt{2x^2+2x+3} - \frac{1}{2} \int \frac{dx}{\sqrt{2(x^2+x+3/2)}}$.
$\boxed{\frac{3}{2}\sqrt{2x^2+2x+3} - \frac{1}{2\sqrt{2}}\log|x+\frac{1}{2}+\sqrt{x^2+x+\frac{3}{2}}| + C}$
Q24
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$\int \frac{dx}{x^2+4x+8}$ equals:
(A) $\frac{1}{2}\tan^{-1}(\frac{x+2}{2}) + C$
(B) $\tan^{-1}(\frac{x+2}{2}) + C$
(C) $\frac{1}{2}\tan^{-1}(x+2) + C$
(D) $\tan^{-1}(x+2) + C$
(A) $\frac{1}{2}\tan^{-1}(\frac{x+2}{2}) + C$
(B) $\tan^{-1}(\frac{x+2}{2}) + C$
(C) $\frac{1}{2}\tan^{-1}(x+2) + C$
(D) $\tan^{-1}(x+2) + C$
$\int \frac{dx}{(x+2)^2+2^2} = \frac{1}{2}\tan^{-1}(\frac{x+2}{2}) + C$.
$\boxed{\text{(A) } \frac{1}{2}\tan^{-1}(\frac{x+2}{2}) + C}$
Q25
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$\int \frac{dx}{\sqrt{2x-x^2}}$ equals:
(A) $\sin^{-1}(x-1) + C$
(B) $\sin^{-1}(1-x) + C$
(C) $\log|x-1+\sqrt{2x-x^2}| + C$
(D) $\tan^{-1}(x-1) + C$
(A) $\sin^{-1}(x-1) + C$
(B) $\sin^{-1}(1-x) + C$
(C) $\log|x-1+\sqrt{2x-x^2}| + C$
(D) $\tan^{-1}(x-1) + C$
$2x-x^2 = -(x^2-2x) = -(x^2-2x+1-1) = 1-(x-1)^2$.
$\int \frac{dx}{\sqrt{1-(x-1)^2}} = \sin^{-1}(x-1) + C$.
$\boxed{\text{(A) } \sin^{-1}(x-1) + C}$