Evaluate: ∫ 1/1+3² x+8² x dx

Correct Answer is : 16tan⁡−11/6 ^{-1}61tan−1 (2tan⁡x3)≤ft(2 x/3)(32tanx) + C

Correct Answer is : 16tan⁡−11/6 ^{-1}61tan−1 (2tan⁡x3)≤ft(2 x/3)(32tanx) + C

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BITSAT 2014 Paper

Section: Mathematics

Question : 111 of 150

Marks: +1, -0

Evaluate: ∫

\frac{1}{1+3\sin^2 x+8\cos^2 x}

Solution:

I = ∫

\frac{1}{1+3\sin^2 x+8\cos^2 x}

Dividing the numerator and denominator by

\cos^2 x

, we get

I = ∫

\frac{\sec^2 x}{sc^2 x+3\tan^2 x+8}

dx = ∫

\frac{\sec^2 x}{4\tan^2 x+9}

Putting tan x = t ⇒

\sec^2 x

dx = dt, we get

I = ∫

\frac{dt}{4t^2+9}

\frac{1}{4} \int \frac{dt}{t^2+\left(\frac{3}{2}\right)^2}

\frac{1}{4} \times \frac{1}{\frac{3}{2}} \tan^{-1} \left(\frac{t}{\frac{3}{2}}\right)

+ C

⇒ I =

\frac{1}{6} \tan^{-1}\left(\frac{2t}{3}\right)

+ C =

\frac{1}{6} \tan^{-1}\left(\frac{2\tan x}{3}\right)

+ C

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