The value of int_0^π/2 dx/(1+tan^3 x) is — JEE Main 1993 Mathematics PYQ

1993

The value of

\int_0^{\pi/2} \dfrac{dx}{1+\tan^3 x}

A.
0
B.
1
C.
$\pi/2$
D.
$\pi/4$

Show answer and solution

Correct option: D

Solution

Use the property I = ∫₀^(π/2) f(x)dx = ∫₀^(π/2) f(π/2 - x)dx.

With f(x) = 1/(1+tan³x), substituting x → π/2-x gives f(π/2-x) = 1/(1+cot³x) = tan³x/(1+tan³x).

Adding the two forms: 2I = ∫₀^(π/2) [1/(1+tan³x) + tan³x/(1+tan³x)] dx = ∫₀^(π/2) 1 dx = π/2.

Therefore I = π/4.

[1993]

Practise Definite Integrals and Areas the way it is asked

A timed set of real JEE Main Definite Integrals and Areas questions, graded as the exam grades. No sign-up.

Start a Definite Integrals and Areas test

More: all JEE Main Definite Integrals and Areas questions · JEE Main 1993 · Mathematics chapter list

JEE Main 1993 Mathematics question on Definite Integrals and Areas