MHTCET 20 Apr 2026 Shift 2 Paper

Question : 129 of 150

Marks: +1, -0

\int \frac{\log x}{(1+\log x)^{2}} d x=

Solution:

Concept:

Identify the integrand as the derivative of a simpler rational function.

Explanation:

Let

y = \frac{x}{1+\log x}

Differentiate

y

with respect to

x

using the quotient rule.

\frac{d}{dx}\left(\frac{x}{1+\log x}\right) = \frac{(1+\log x)(1) - x\left(\frac{1}{x}\right)}{(1+\log x)^2}

Since

x \cdot \frac{1}{x} = 1

, the numerator becomes

1+\log x - 1 = \log x

Thus,

\frac{d}{dx}\left(\frac{x}{1+\log x}\right) = \frac{\log x}{(1+\log x)^2}

This exactly matches the integrand.

Therefore, the integral is

y + c

Answer:

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

Correct option: Option C.

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