Evaluate $\int_{0}^{\pi / 2} \frac{1}{1+(\tan x)^{2 / 3}} d x$

CBSE Class 12 · 2022 · Definite IntegralsShort Answer

Evaluate

\int_{0}^{π /2} \frac{1}{1 + ( t a n x ) ^{2/3}} d x

Answer: $\frac{π}{4}$

Let

I = \int_{0}^{π /2} \frac{d x}{1 + ( tan x ) ^{2/3}}

. Apply

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

with

a = \frac{π}{2}

, using

tan (\frac{π}{2} - x) = cot x

I = \int_{0}^{π /2} \frac{d x}{1 + ( cot x ) ^{2/3}} = \int_{0}^{π /2} \frac{( tan x ) ^{2/3}}{1 + ( tan x ) ^{2/3}} d x

Adding the two expressions for

I

2 I = \int_{0}^{π /2} \frac{1 + ( tan x ) ^{2/3}}{1 + ( tan x ) ^{2/3}} d x = \int_{0}^{π /2} d x = \frac{π}{2}

Hence

I = \frac{π}{4}

.
(This value is independent of the exponent; the same reflection gives

\frac{π}{4}

for any power of

tan x

CBSE Class 12 Board · 2022 · Subjective