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AP EAMCET Engineering 2018 Apr 22 Shift 1 Paper

Section: Mathematics

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Question : 72 of 160

Marks: +1, -0

∫

\frac{\sin 2x\,dx}{\sin^{4} x + \cos^{4} x}

\tan^{-1}

(f(x)) + c , then f

\left(\frac{\pi}{3}\right)

=

1
2
3
4

Solution:

Let.

I = \int \frac{\sin 2x\,dx}{\sin^{4} x + \cos^{4} x}

= \int \frac{2\sin x + \cos x}{\sin^{4} x + \cos^{4} x} \, dx

Dividing by

\cos^{2} x

in numerator and denominator

= \int \frac{2\tan x \sec^{2} x}{1 + (\tan^{2} x)^{2}} \, dx

Again let

\tan^{2} x = t

I = \int \frac{dt}{1 + t^{2}}

= \tan^{-1} t + C

\tan^{-1}(\tan^{3} x) + C

By comparing with

\tan^{-1} f(x) + C.

we get

f(x) = \tan^{2} x

f\left(\frac{\pi}{3}\right) = \left\{\tan\left(\frac{\pi}{3}\right)\right\}^{2}

= (\sqrt{3})^{2} = 3

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