Evaluate: $\int_{0}^{\pi} \frac{4 x \sin x}{1+\cos ^{2} x}\, dx$

CBSE Class 12 · 2014 · Definite IntegralsLong Answer

Evaluate:

\int_{0}^{π} \frac{4 x s i n x}{1 + c o s ^{2} x} d x

Answer: $π^{2}$

Let

I = \int_{0}^{π} \frac{4 x sin x}{1 + cos ^{2} x} d x

. Apply

x \to π - x

and add:

2 I = 4 π \int_{0}^{π} \frac{sin x}{1 + cos ^{2} x} d x

Put

t = cos x

(

x = 0 \to t = 1

x = π \to t = - 1

2 I = 4 π \int_{- 1}^{1} \frac{d t}{1 + t ^{2}} = 4 π [tan^{- 1} t]_{- 1}^{1} = 4 π \cdot \frac{π}{2} = 2 π^{2}

Hence

I = π^{2}

CBSE Class 12 Board · 2014 · Subjective