Evaluate: (x+1) (x+2)(x+3) d x - Vidyadip

Question

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

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Answer

$\begin{aligned} & \text { Let } I =\int \frac{x+1}{(x+2)(x+3)} d x \\ & \frac{x+1}{(x+2)(x+3)}=\frac{A}{x+2}+\frac{B}{x+3} \\ & x+1=A(x+3)+B(x+2) \ldots \ldots . . .(1)\end{aligned}$
∴ Putting x = -2 in equation (i) we get
-1 = A
∴ A = -1
∴ Putting x = -3 in equation (i) we get
-2 = -B
∴ B = 2
∴(x+1)/((x+2)(x+3))=1/(x+2)+2/(x+3)
$\begin{aligned} & \therefore I=\int\left[-\frac{1}{x+2}+\frac{2}{x+3}\right] d x \\ & \therefore I=-\log |x+2|+2 \log |x+3|+c\end{aligned}$

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