First slide

Methods of integration

Question

$\int \frac{(x + 1)^{2} d x}{x (x^{2} + 1)} is equal to$

Moderate

A

$\log_{e} x + c$
B

$\log_{e} x + 2 \tan^{- 1} x + c$
C

$\log_{e} \frac{1}{x^{2} + 1} + c$
D

$\log_{e} \{x (x^{2} + 1)\} + c$

Solution

$\begin{array}{l} \int \frac{(x + 1)^{2}}{x (x^{2} + 1)} dx = \int \frac{x^{2} + 1 + 2 x}{x (x^{2} + 1)} dx \\ = \int \frac{x^{2} + 1}{x (x^{2} + 1)} dx + 2 \int \frac{x}{x (x^{2} + 1)} dx \\ = \int \frac{dx}{x} + 2 \int \frac{dx}{x^{2} + 1} = \log_{e} x + 2 \tan^{- 1} x + c \end{array}$

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