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Question: Answered & Verified by Expert

Let Ix=x+7x dx and I9=12+7loge7. If I1=α+7loge1+22, then α4 is equal to _____.

MathematicsIndefinite IntegrationJEE MainJEE Main 2023 (12 Apr Shift 1)
Solution:
2534 Upvotes Verified Answer
The correct answer is: 64

Given,

Ix=x+7x dx

Now let, x+7x=t2-7x2dx=2t dt

dx=-14tt2-12dt

So, Ix=-14t2t2-12dt

Ix=-14dtt2+1t2-2

Ix=-1421-1t2t+1t2-4+1+1t2t-1t2dt

Ix=-714lnt+1t-2t+1t+2-1t-1t+c

Now when x=9, t=43

I9= 12+7×ln7=-74ln172+7×127+c

 c=72ln7

Now when x=1, t=22

 I1=+74ln22+122-12+7×227+72ln7

I1=72ln22+127+22+72ln7

I1=7 ln22+1-72ln7+22+72ln7

 α=22α4=64

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