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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.15 Question 6

Answers (1)

Answer:

$\log \left|\frac{(x+3)^{2}}{(x+2)}\right|+C$

Hint:

To solve this problem use special integration formula

Given:

$\int \frac{(x+1)}{(x+2)(x+3)} d x$

Solution:

Let $I=\int \frac{(x+1)}{(x+2)(x+3)} d x$

Applying partial fraction

$\begin{aligned} &\frac{(x+1)}{(x+2)(x+3)}=\frac{A}{(x+2)}+\frac{B}{(x+3)} \\ & \end{aligned}$

$(x+1)=A(x+3)+B(x+2)$

Putting

$\begin{aligned} &x=-2 \\ \end{aligned}$

$(-2+1)=A(-2+3)+B(-2+2) \\$

$A=-1$

Putting

$\begin{aligned} &x=-3 \\ \end{aligned}$

$(-3+1)=A(-3+3)+B(-3+2) \\$

$-2=0+B(-1) \\$

$B=2$

$\begin{aligned} &I=\int \frac{x+1}{(x+1)(x+2)} d x=\int \frac{-1}{x+2} d x+\int \frac{2}{x+3} d x \\ \end{aligned}$

$=-\log |x+2|+2 \log |x+3| \\$

$=\log \left|\frac{(x+3)^{2}}{(x+2)}\right|+C$

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