IIT JEE 1983 Maths - MCQ Question 1

Calculus Level 3

∫ 0 π / 2 cot ⁡ x cot ⁡ x + tan ⁡ x d x = ? \large \int_0^{\pi/2}\dfrac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} \, dx = \, ?

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π / 4 \pi/4 π / 2 \pi/2 π \pi None of the others

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Shubhamkar Ayare by Shubhamkar Ayare , 22, India

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2 solutions

Chew-Seong Cheong
Dec 28, 2016

Relevant wiki: Integration Tricks

Easier solution after getting an idea from Nivedit Jain

I = ∫ 0 π 2 cot ⁡ x cot ⁡ x + tan ⁡ x d x Dividing the integrand up and down by cot ⁡ = ∫ 0 π 2 1 1 + tan ⁡ x d x Using identity: ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x = 1 2 ∫ 0 π 2 1 1 + tan ⁡ x + 1 1 + cot ⁡ x d x = 1 2 ∫ 0 π 2 1 1 + tan ⁡ x + tan ⁡ x tan ⁡ x + 1 d x = 1 2 ∫ 0 π 2 1 + tan ⁡ x 1 + tan ⁡ x d x = 1 2 x ∣ 0 π 2 = π 4 \begin{aligned} I & = \int_0^\frac \pi 2 \frac {\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} dx & \small \color{#3D99F6} \text{Dividing the integrand up and down by} \sqrt{\cot} \\ & = \int_0^\frac \pi 2 \frac 1{1+\tan x} dx & \small \color{#3D99F6} \text{Using identity: }\int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx \\ & = \frac 12 \int_0^\frac \pi 2 \frac 1{1+\tan x} + \frac 1{1+\cot x} dx \\ & = \frac 12 \int_0^\frac \pi 2 \frac 1{1+\tan x} + \frac {\tan x}{\tan x+1} dx \\ & = \frac 12 \int_0^\frac \pi 2 \frac {1+\tan x}{1+\tan x} dx \\ & = \frac 12 x \bigg|_0^\frac \pi 2 = \boxed{\dfrac \pi 4} \end{aligned}

Previous solution

I = ∫ 0 π 2 cot ⁡ x cot ⁡ x + tan ⁡ x d x Using identity: ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x = 1 2 ∫ 0 π 2 cot ⁡ x cot ⁡ x + tan ⁡ x + cot ⁡ ( π 2 − x ) cot ⁡ ( π 2 − x ) + tan ⁡ ( π 2 − x ) d x = 1 2 ∫ 0 π 2 cot ⁡ x cot ⁡ x + tan ⁡ x + tan ⁡ x tan ⁡ x + cot ⁡ x = 1 2 ∫ 0 π 2 1 d x = π 4 \begin{aligned} I & = \int_0^\frac \pi 2 \frac {\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} dx & \small \color{#3D99F6} \text{Using identity: }\int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx \\ & = \frac 12 \int_0^\frac \pi 2 \frac {\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} + \frac {\sqrt{\cot \left(\frac \pi 2 - x \right)}}{\sqrt{\cot \left(\frac \pi 2 - x \right)}+\sqrt{\tan \left(\frac \pi 2 - x \right)}} dx \\ & = \frac 12 \int_0^\frac \pi 2 \frac {\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} + \frac {\sqrt{\tan x}}{\sqrt{\tan x}+\sqrt{\cot x}} \\ & = \frac 12 \int_0^\frac \pi 2 1 \ dx \\ & = \boxed{\dfrac \pi 4} \end{aligned}

9 Helpful 0 Interesting 0 Brilliant 0 Confused

nice one but i have a better way see my solution. But learnt a lot of new things from your solutions as always thanks keep posting solutions and help all of us

Nivedit Jain - 4 years, 2 months ago

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Thanks, you give me an idea. I am changing it to a better solution.

Chew-Seong Cheong - 4 years, 2 months ago

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sir try this one out https://brilliant.org/problems/waves-new-style/

Nivedit Jain - 4 years, 2 months ago
Nivedit Jain
Mar 26, 2017

1 Helpful 0 Interesting 0 Brilliant 0 Confused

@Nivedit Jain how did you replace ( (1-tan(x) (1+(tan(x)) with Sec^2(x) ????

Sälmän Rähmän - 3 years, 10 months ago

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sorry my mistake

Nivedit Jain - 3 years, 9 months ago

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