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The value of $I = \int_{0}^{π} \frac{x d x}{4 \cos^{2} x + 9 \sin^{2} x}$ is

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a

π2/12

b

π2/4

c

π2/6

d

π2/3

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detailed solution

Correct option is A

Using Property 9, we haveI=∫0π (π−x)dx4cos2⁡(π−x)+9sin2⁡(π−x)=π∫0π dx4cos2⁡x+9sin2⁡x−∫0π xdx4cos2⁡x+9sin2⁡x2I=π∫0π dx4cos2⁡x+9sin2⁡x=2π∫0π/2 dx4cos2⁡x+9sin2⁡x=2π∫0π/2 sec2⁡xdx4+9tan2⁡x=2π∫0∞ dt4+9t2 (t=tan⁡x)=2π9∫0∞ dtt2+4/9=2π9⋅32tan−1⁡32t0∞=π26Hence I=π2/12

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