Exercise 7.5 Practice
Integration by Partial Fractions – NCERT Solutions
Q1
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Integrate the function: $\frac{1}{(x+2)(x+3)}$
Let $\frac{1}{(x+2)(x+3)} = \frac{A}{x+2} + \frac{B}{x+3}$.
$1 = A(x+3) + B(x+2)$.
Put $x=-2 \Rightarrow A=1$. Put $x=-3 \Rightarrow B=-1$.
$\boxed{\log|\frac{x+2}{x+3}| + C}$
Q2
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Integrate the function: $\frac{2x}{x^2-4}$
Let $x^2-4 = t \Rightarrow 2x dx = dt$.
$\boxed{\log|x^2-4| + C}$
Q3
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Integrate the function: $\frac{1}{x^2-x-2}$
Factor denominator: $(x-2)(x+1)$.
$\frac{1}{(x-2)(x+1)} = \frac{1}{3}(\frac{1}{x-2} - \frac{1}{x+1})$.
$\boxed{\frac{1}{3}\log|\frac{x-2}{x+1}| + C}$
Q4
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Integrate the function: $\frac{3x}{(x-2)(x+1)}$
$\frac{A}{x-2} + \frac{B}{x+1}$. $3x = A(x+1) + B(x-2)$.
$x=2 \Rightarrow 6=3A \Rightarrow A=2$. $x=-1 \Rightarrow -3=-3B \Rightarrow B=1$.
$\boxed{2\log|x-2| + \log|x+1| + C}$
Q5
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Integrate the function: $\frac{x}{(x+1)(x+2)(x+3)}$
$\frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{x+3}$.
$A=-1/2, B=2, C=-3/2$.
$\boxed{-\frac{1}{2}\log|x+1| + 2\log|x+2| - \frac{3}{2}\log|x+3| + C}$
Q6
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Integrate the function: $\frac{2x-1}{(x-1)^2}$
$\frac{A}{x-1} + \frac{B}{(x-1)^2}$. $2x-1 = A(x-1) + B$.
$x=1 \Rightarrow B=1$. Coeff $x \Rightarrow A=2$.
$\boxed{2\log|x-1| - \frac{1}{x-1} + C}$
Q7
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Integrate the function: $\frac{x^2+1}{x^2-4}$
Improper fraction. Divide: $1 + \frac{5}{x^2-4}$.
$\int dx + 5 \int \frac{dx}{x^2-2^2}$.
$\boxed{x + \frac{5}{4}\log|\frac{x-2}{x+2}| + C}$
Q8
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Integrate the function: $\frac{1}{x(x^2+1)}$
$\frac{A}{x} + \frac{Bx+C}{x^2+1}$. $1 = A(x^2+1) + (Bx+C)x$.
$A=1, B=-1, C=0$.
$\boxed{\log|x| - \frac{1}{2}\log(x^2+1) + C}$
Q9
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Integrate the function: $\frac{x}{(x-1)(x^2+1)}$
$\frac{A}{x-1} + \frac{Bx+C}{x^2+1}$. $x = A(x^2+1) + (Bx+C)(x-1)$.
$A=1/2, B=-1/2, C=1/2$.
$\boxed{\frac{1}{2}\log|x-1| - \frac{1}{4}\log(x^2+1) + \frac{1}{2}\tan^{-1}x + C}$
Q10
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Integrate the function: $\frac{2x}{(x^2+1)(x^2+2)}$
Let $x^2=t \Rightarrow 2x dx = dt$.
$\int \frac{dt}{(t+1)(t+2)} = \int (\frac{1}{t+1} - \frac{1}{t+2}) dt$.
$\boxed{\log|\frac{x^2+1}{x^2+2}| + C}$
Q11
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Integrate the function: $\frac{1}{e^x+1}$
$\int \frac{e^{-x}}{1+e^{-x}} dx$. Let $1+e^{-x}=t$.
$\boxed{x - \log(e^x+1) + C}$
Q12
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Integrate the function: $\frac{x^2}{(x-1)(x-2)}$
Improper. $x^2 = (x^2-3x+2) + 3x-2$.
$1 + \frac{3x-2}{(x-1)(x-2)} = 1 + \frac{-1}{x-1} + \frac{4}{x-2}$.
$\boxed{x - \log|x-1| + 4\log|x-2| + C}$
Q13
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Integrate the function: $\frac{1}{x(x^5+1)}$
Multiply by $x^4$. Let $x^5=t$.
$\boxed{\frac{1}{5}\log|\frac{x^5}{x^5+1}| + C}$
Q14
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Integrate the function: $\frac{\sin x}{(1+\cos x)(2+\cos x)}$
Let $\cos x = t \Rightarrow -\sin x dx = dt$.
$\boxed{-\log|\frac{1+\cos x}{2+\cos x}| + C}$
Q15
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Integrate the function: $\frac{x^2+x+1}{(x+2)(x^2+1)}$
$\frac{A}{x+2} + \frac{Bx+C}{x^2+1}$.
$A=3/5, B=2/5, C=1/5$.
$\boxed{\frac{3}{5}\log|x+2| + \frac{1}{5}\log(x^2+1) + \frac{1}{5}\tan^{-1}x + C}$
Q16
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Integrate the function: $\frac{1}{x^3-1}$
$\frac{1}{(x-1)(x^2+x+1)} = \frac{1/3}{x-1} - \frac{1/3(x+2)}{x^2+x+1}$.
Split second term into derivative and constant.
$\boxed{\frac{1}{3}\log|x-1| - \frac{1}{6}\log(x^2+x+1) - \frac{1}{\sqrt{3}}\tan^{-1}(\frac{2x+1}{\sqrt{3}}) + C}$
Q17
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Integrate the function: $\frac{2x+1}{(x+2)(x-3)}$
$\frac{A}{x+2} + \frac{B}{x-3}$.
$A=3/5, B=7/5$.
$\boxed{\frac{3}{5}\log|x+2| + \frac{7}{5}\log|x-3| + C}$
Q18
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Integrate the function: $\frac{x^2}{(x^2+1)(x^2+4)}$
Let $x^2=y$ for partial fraction. $\frac{y}{(y+1)(y+4)} = \frac{-1/3}{y+1} + \frac{4/3}{y+4}$.
$\boxed{-\frac{1}{3}\tan^{-1}x + \frac{2}{3}\tan^{-1}(\frac{x}{2}) + C}$
Q19
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Integrate the function: $\frac{1}{x(x^3-1)}$
Multiply by $x^2$. Let $x^3=t$.
$\boxed{\frac{1}{3}\log|\frac{x^3-1}{x^3}| + C}$
Q20
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Integrate the function: $\frac{3x-2}{(x+1)^2(x+3)}$
$\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x+3}$.
$A=11/4, B=-5/2, C=-11/4$.
$\boxed{\frac{11}{4}\log|\frac{x+1}{x+3}| + \frac{5}{2(x+1)} + C}$
Q21
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Integrate the function: $\frac{1}{e^{2x}-1}$
$\frac{e^{-2x}}{1-e^{-2x}}$. Let $1-e^{-2x}=t$.
$\boxed{\frac{1}{2}\log|1-e^{-2x}| + C}$
Q22
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$\int \frac{dx}{x(x+1)}$ equals:
(A) $\log|\frac{x}{x+1}| + C$
(B) $\log|\frac{x+1}{x}| + C$
(C) $\log|x(x+1)| + C$
(D) $\log|x| + \log|x+1| + C$
(A) $\log|\frac{x}{x+1}| + C$
(B) $\log|\frac{x+1}{x}| + C$
(C) $\log|x(x+1)| + C$
(D) $\log|x| + \log|x+1| + C$
$\frac{1}{x} - \frac{1}{x+1}$.
$\boxed{\text{(A) } \log|\frac{x}{x+1}| + C}$
Q23
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$\int \frac{x}{x^2-1} dx$ equals:
(A) $\log|x^2-1| + C$
(B) $\frac{1}{2}\log|x^2-1| + C$
(C) $2\log|x^2-1| + C$
(D) $\log|\frac{x-1}{x+1}| + C$
(A) $\log|x^2-1| + C$
(B) $\frac{1}{2}\log|x^2-1| + C$
(C) $2\log|x^2-1| + C$
(D) $\log|\frac{x-1}{x+1}| + C$
Let $x^2-1=t \Rightarrow 2x dx = dt$.
$\boxed{\text{(B) } \frac{1}{2}\log|x^2-1| + C}$