What is (x²) dx equal to?

Correct Answer is : 2x In (x) – 2x + c

{2\;{In}\;{left}( x {right})}{x} - 2x + c

Correct Answer is : 2x In (x) – 2x + c

Test Index

UPSC NDA I 2019 Mathematics Solved Paper

Question : 87 of 120

Marks: +1, -0

What is

\int \ln(x^2) dx

equal to?

Solution:

Concept:

Integration by parts and the logarithmic property

\ln(x^2)=2\ln x

Explanation:

First, simplify the integral using

\ln(x^2)=2\ln x

Thus,

\int \ln(x^2) \,dx = \int 2\ln x \,dx = 2\int \ln x\,dx

Now, integrate

\int \ln x\,dx

using integration by parts with

u=\ln x

and

dv=dx

Then

du=\frac{1}{x}dx

and

v=x

Apply the formula:

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

\int \ln x\,dx = x\ln x - \int x\cdot\frac{1}{x}dx = x\ln x - \int 1\,dx = x\ln x - x + c

Multiply by 2:

2(x\ln x - x) + c = 2x\ln x - 2x + c

Therefore,

\int \ln(x^2)\,dx = 2x\ln x - 2x + c

Answer:

2x\ln x - 2x + c

(Option A)

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