Evaluate: $\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}} d x$

CBSE Class 12 · 2024 · Definite IntegralsShort Answer

Evaluate:

\int_{0}^{π} \frac{e ^{c o s x}}{e ^{c o s x} + e ^{- c o s x}} d x

Answer: $\frac{π}{2}$

Let

I = \int_{0}^{π} \frac{e ^{c o s x}}{e ^{c o s x} + e ^{- c o s x}} d x

. Apply

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

with

a = π

; since

cos (π - x) = - cos x

I = \int_{0}^{π} \frac{e ^{- c o s x}}{e ^{- c o s x} + e ^{c o s x}} d x

Adding the two forms:

2 I = \int_{0}^{π} \frac{e ^{c o s x} + e ^{- c o s x}}{e ^{c o s x} + e ^{- c o s x}} d x = \int_{0}^{π} d x = π

Hence

I = \frac{π}{2}

CBSE Class 12 Board · 2024 · Subjective