JEE Main 2023MathematicsIndefinite IntegrationMediumMCQ

JEE Main 2023 — Indefinite Integration Question with Solution

JEE Main 2023 (08 Apr Shift 1)

Question

Let Ix=∫x+1x1+xex2dx, x>0. If limx→∞Ix=0 then I1 is equal to

Choose an option

Show full solutionCorrect option: A
Correct answer
Ae+2e+1-logee+1

Step-by-step explanation

Given,

Ix=∫x+1x1+xex2dx

Now let 1+xex=t

⇒exx+1dx=dt

So, Ix=∫1t-1t2dt

⇒Ix=∫1-t2+t2t-1t2dt

⇒Ix=∫-t+1t2+1t-1dt

⇒Ix=∫-1t-1t2+1t-1dt

⇒Ix=-lnt+1t+lnt-1+C

⇒Ix=lnxexxex+1+1xex+1+C

Also given,limx→∞Ix=0

⇒limx→∞Ix=limx→∞ln1-1xex+1+1xex+1+C

⇒limx→∞Ix=ln1-0+0+C

⇒C=0

Now finding,

I1=lnee+1+1e+1=1+1e+1-ln(e+1)

⇒I1=e+2e+1-lne+1

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About this question

This is a previous-year question from JEE Main 2023, covering the Indefinite 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.