If ' n ' is a natural number, then { ⁿ x}{ ⁿ+2} x} d x=

Correct Answer is : tan⁡n+1xn+1+C{ ⁿ+1} x}{n+1}+Cn+1tann+1x+C

Correct Answer is : tan⁡n+1xn+1+C{ ⁿ+1} x}{n+1}+Cn+1tann+1x+C

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KCET 2026 Mathematics Solved Paper

Question : 39 of 60

Marks: +1, -0

If '

n

\int \frac{\sin ^{n} x}{\cos ^{n+2} x} d x=

Solution:

Concept:

Rewrite the integrand as

\tan^n x \sec^2 x

and use substitution.

Explanation:

Given integral:

I = \int \frac{\sin^n x}{\cos^{n+2} x} \, dx

Split the powers of

\cos x

\frac{\sin^n x}{\cos^{n+2} x} = \frac{\sin^n x}{\cos^n x} \cdot \frac{1}{\cos^2 x}

Use

\frac{\sin x}{\cos x} = \tan x

and

\frac{1}{\cos^2 x} = \sec^2 x

\frac{\sin^n x}{\cos^{n+2} x} = \tan^n x \sec^2 x

Therefore,

I = \int \tan^n x \sec^2 x \, dx

Let

t = \tan x

. Then

dt = \sec^2 x \, dx

So,

I = \int t^n \, dt = \frac{t^{n+1}}{n+1} + C

Substitute back

t = \tan x

I = \frac{\tan^{n+1} x}{n+1} + C

Answer:

Option D:

\frac{\tan^{n+1} x}{n+1} + C

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