COMEDK 2025 Morning Solved Paper

Question : 152 of 180

Marks: +1, -0

\int \log x^2 d x=

Solution:

Concept:

Integration of the logarithmic function using the method of integration by parts.

Explanation:

First, rewrite the integrand:

\log x^2 = 2 \log x

(taking

\log

as natural logarithm).

So the integral becomes:

\int \log x^2 \, dx = 2 \int \log x \, dx

Now integrate

\int \log x \, dx

using integration by parts.

Let

u = \log x

and

dv = dx

Then

du = \frac{1}{x} dx

and

v = x

Using the formula

\int u \, dv = uv - \int v \, du

, we get:

\int \log x \, dx = x \log x - \int x \cdot \frac{1}{x} dx = x \log x - \int 1 \, dx = x \log x - x + C

Multiply by 2:

2(x \log x - x) = 2x \log x - 2x + C

Express

2x \log x

x \log x^2

x \log x^2 - 2x + C

Answer:

Option D:

x \log x^2 - 2x + c

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