What is dx/x(x² + 1) equal to?

Correct Answer is : 12ln⁡(x2x2+1)+C1/2 ≤ft(x²/x² + 1) + C21ln(x2+1x2)+C

Correct Answer is : 12ln⁡(x2x2+1)+C1/2 ≤ft(x²/x² + 1) + C21ln(x2+1x2)+C

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UPSC NDA 2 2021 Mathematics Solved Paper

Question : 93 of 120

Marks: +1, -0

What is

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

equal to?

Solution:

Concept:

Use partial fraction decomposition to split the integrand into simpler terms for direct integration.

Explanation:

Rewrite

\frac{1}{x(x^2+1)}

\frac{1}{x} - \frac{x}{x^2+1}

Integrate term by term:

\int \frac{dx}{x} = \ln|x| + C_1

For

\int \frac{x}{x^2+1}\,dx

, substitute

u = x^2+1

, so

du = 2x\,dx

Then

\int \frac{x}{x^2+1}\,dx = \frac{1}{2}\int \frac{du}{u} = \frac{1}{2}\ln|u| + C_2 = \frac{1}{2}\ln|x^2+1| + C_2

Thus the whole integral is

\ln|x| - \frac{1}{2}\ln|x^2+1| + C

Write

\ln|x|

\frac{1}{2}\ln x^2

Combine:

\frac{1}{2}\ln x^2 - \frac{1}{2}\ln(x^2+1) = \frac{1}{2}\ln\left(\frac{x^2}{x^2+1}\right) + C

Answer:

\frac{1}{2} \ln\left(\frac{x^2}{x^2 + 1}\right) + C

(option A).

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