The value of {n {{cosec}² xⁿ-1}}{x^{(1-n)}} d x is ...

Correct Answer is : log⁡(sin⁡xn)+c ≤ft( xⁿ)+clog(sinxn)+c

Correct Answer is : log⁡(sin⁡xn)+c ≤ft( xⁿ)+clog(sinxn)+c

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MHTCET 18 Apr 2026 Shift 1 Paper

Question : 130 of 150

Marks: +1, -0

The value of

\int \frac{n \sqrt{\operatorname{cosec}^{2} x^{n}-1}}{x^{(1-n)}} d x

is ...

Solution:

Concept:

Use the identity

\operatorname{cosec}^{2}\theta - 1 = \cot^{2}\theta

and then integrate by substitution.

Explanation:

Start with the integral:

\int \frac{n\sqrt{\operatorname{cosec}^{2}x^{n}-1}}{x^{1-n}}\,dx.

Using

\operatorname{cosec}^{2}x^{n}-1 = \cot^{2}x^{n}

, the square root becomes

\left|\cot x^{n}\right|

For the standard integral, we take the positive value, so the integrand is:

\frac{n\cot x^{n}}{x^{1-n}} = n x^{n-1}\cot x^{n}.

Now substitute:

t = x^{n} \Rightarrow dt = n x^{n-1}\,dx.

The integral becomes:

\int \cot t\,dt = \log|\sin t| + C.

Replace

t

with

x^{n}

\log\left|\sin\left(x^{n}\right)\right| + C.

This matches the form

\log\left(\sin x^{n}\right) + c

Answer:

Option B:

\log\left(\sin x^{n}\right) + c

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