JEE Main 2021MathematicsDefinite IntegrationMediumMCQ

JEE Main 2021 — Definite Integration Question with Solution

JEE Main 2021 (27 Jul Shift 1)

Question

The value of the definite integral ∫-π4π4dx1+excosxsin4x+cos4x is equal to :

Choose an option

Show full solutionCorrect option: B
Correct answer
Bπ22

Step-by-step explanation

I=∫-π4π4dx1+excosxsin4x+cos4x    …1

Using ∫abfxdx=∫abfa+b-xdx

I=∫-π/4π/4dx1+e-xcosxsin4x+cos4x    ...2

Add 1 and 2

2I=∫-π4π4dxsin4x+cos4x

⇒2I=2∫0π4dxsin4x+cos4x

⇒I=∫0π41+tan2xsec2xtan4x+1dx

⇒I=∫0π41+1tan2xsec2xtanx-1tanx2+2dx

Put tanx-1tanx=t

⇒1+1tan2xsec2xdx=dt

So, I=∫-∞0dtt2+2=12tan-1t2-∞0

⇒I=0-12-π2=π22

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About this question

This is a previous-year question from JEE Main 2021, covering the Definite Integration chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.