$\int \frac{dx}{x\left(x^2+1\right)}$

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Indefinite Integration

Question:

$\int \frac{dx}{x\left(x^2+1\right)}$ equals

Options:

$\log\left|x\right|+\frac{1}{2}\log \left(x^2+1\right)+C$

$-\log\left|x\right|+\frac{1}{2}\log \left(x^2+1\right)+C$

$\log\left|x\right|-\frac{1}{2}\log \left(x^2+1\right)+C$

None

Correct Answer:

$\log\left|x\right|-\frac{1}{2}\log \left(x^2+1\right)+C$

Explanation:

$I = \int{\frac{dx}{x(x^2+ 1)}}=\int\frac{x^{-3}}{1+x^{-2}}dx$

$1+x^{-2}=y$

$dy=-2x^{-3}dx$

so $I=-\frac{1}{2}\int\frac{dy}{y}=-\frac{1}{2}\log(y)+C$

$=-\frac{1}{2}\log(1+x^{-2})+C$

$=\log x-\frac{1}{2}\log(x^2+1)+C$