Iₙ= ⁿ x \, dx, then 6 I₆-5 I₄=

Correct Answer is : cos⁡5xsin⁡x^{5} x xcos5xsinx

Correct Answer is : cos⁡5xsin⁡x^{5} x xcos5xsinx

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TS EAMCET 5 May 2018 Shift 2 Solved Paper

Section: Mathematics

Question : 74 of 160

Marks: +1, -0

I_{n}=\int \cos^{n} x \, dx,

6 I_{6}-5 I_{4}=

Solution:

Consider the integral.

I_{n}=\int \cos^{n} x \, dx

The above integral is solved as,

I_{n}=\int \cos^{n-1} x \cos^{n} x \, dx

=\cos^{n-1} x \sin x+\int \cos^{n-2} x

(n-1) \sin^{2} x \, dx

=\cos^{n-1} x \sin x+(n-1)

\int \cos^{n-2} x-(n-1)\int \cos^{n} x \, dx

=\cos^{n-1} x \sin x+(n-1)

I_{n-2}-(n-1) I_{n}

Solve further,

I_{n}(1+n-1)=\cos^{n-1} x \sin x

+(n-1) I_{n-2}

n I_{n}-(n-1) I_{n-2}

=\cos^{n-1} x \sin x

Therefore,

6 I_{6}-5 I_{4}=\cos^{5} x \sin x

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