Evaluate: $\int_{0}^{\pi} \frac{x}{1+\sin x} d x$

CBSE Class 12 · 2010 · Definite IntegralsLong Answer

Evaluate:

\int_{0}^{π} \frac{x}{1 + s i n x} d x

Answer: $π$

Let

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

. Apply

x \to π - x

(with

sin (π - x) = sin x

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

Multiply numerator and denominator by

(1 - sin x)

\frac{1 - sin x}{cos ^{2} x} = sec^{2} x - sec x tan x

2 I = π [tan x - sec x]_{0}^{π} = π [(0 + 1) - (0 - 1)] = 2 π

Hence

I = π

CBSE Class 12 Board · 2010 · Subjective