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JEE Advanced · 2019 · 27 May 2019 · Paper 2 · Definite Integrals

The value of the integral

\int_{0}^{π /2} \frac{3 c o s θ}{( c o s θ + s i n θ ) ^{5}} d θ

equals

Answer: 0.5

Solution

Let

I = \int_{0}^{π /2} \frac{3 c o s θ}{( c o s θ + s i n θ ) ^{5}} d θ

. Using the property

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

, we get

I = \int_{0}^{π /2} \frac{3 s i n θ}{( c o s θ + s i n θ ) ^{5}} d θ

. Adding:

2 I = 3 \int_{0}^{π /2} \frac{d θ}{( c o s θ + s i n θ ) ^{4}} = 3 \int_{0}^{π /2} \frac{s e c ^{2} θ d θ}{( 1 + t a n θ ) ^{4}}

. Substituting

1 + tan θ = t

:

2 I = 6 \int_{1}^{\infty} (t^{- 3} - t^{- 4}) d t = 6 [\frac{1}{2} - \frac{1}{3}] = 1

. Thus

I = 0.50

.

JEE Advanced · 2019 · 27 May 2019 · Paper 2 · Numerical answer

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