   Chapter 7.R, Problem 31E

Chapter
Section
Textbook Problem

# 1–40 Evaluate the integral. ∫ 0 In10 e x e x − 1 e x + 8 d x

To determine

To Find:0ln10exex1ex+8dx

Explanation

Formula Used:dxx2+a2=1atan1(xa)+C

Calculation:

Here the integrand looks weird, but the integral is easy to evaluate just by using a suitable substitution. The substitution, we will be using, is ex1=t2. Basically we use the substitution which makes the integrand simpler. Now, differentiate ex1=t2 with respect to x to get

exdx=2tdt

Now, find the limit of integration for variable t as shown below:

When x=0,t=0

When x=ln10;ex1=9=t2,t=3

After substitution, the definite integral reduces to

0ln10exex1dxex+8=03t2×2tdt1+t2+8=2

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