Prove that $\int_{0}^{\pi / 4}(\sqrt{\tan x}+\sqrt{\cot x}) d x=\sqrt{2} \cdot \frac{\pi}{2}$

CBSE Class 12 · 2012 · Definite IntegralsLong Answer

Prove that

\int_{0}^{π /4} (tan x + cot x) d x = 2 \cdot \frac{π}{2}

Answer: Proved

tan x + cot x = \frac{sin x + cos x}{sin x cos x} = 2 \cdot \frac{sin x + cos x}{2 sin x cos x} = 2 \cdot \frac{sin x + cos x}{1 - ( sin x - cos x ) ^{2}}

.
Put

z = sin x - cos x \Rightarrow d z = (cos x + sin x) d x

;

x = 0 \to z = - 1

x = \frac{π}{4} \to z = 0

LHS = 2 \int_{- 1}^{0} \frac{d z}{1 - z ^{2}} = 2 [sin^{- 1} z]_{- 1}^{0} = 2 [0 - (- \frac{π}{2})] = 2 \cdot \frac{π}{2}

Hence proved.

CBSE Class 12 Board · 2012 · Subjective