Evaluate the indefinite integral. \int \frac{e^{t}dt}{e^{2t}+2e^{t}+1}

Patricia Crane 2021-12-12 Answered
Evaluate the indefinite integral.
etdte2t+2et+1
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Expert Answer

Shannon Hodgkinson
Answered 2021-12-13 Author has 34 answers

Step 1
Given: I=etdte2t+2et+1...(1)
for evaluating given integral we substitute
et=x...(2)
now differentiating equation (2) with respect to t
so,
ddt(et)=ddt(x)   (ddx(ex)=ex)
et=dxdt
etdt=dx
now replacing etdt ask with dxandet with x in equation (1)
Step 2
so,
I=dxx2+2x+1   (a2+2ab+b2=(a+b)2)
=dxx2+2(x)(1)+12
=dx(x+1)2
=(x+1)2dx   ((x+a)ndx=(x+a)n+1n+1+c)
=(x+1)2+12+1+c
=(x+1)11+c
=1x+1+c...(3)
Step 3
now replacing x with et in equation (3)
I=1et+1+c
hence, given integral is equal to 1et+1+c.

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Andrew Reyes
Answered 2021-12-14 Author has 24 answers

We have:
ete2t+2et+1dt
1u2+2u+1du
1(u+1)2du
1v2dv
Evaluate the integral
1v
1u+1
Substitute back
1et+1
Answer:
1et+1+C

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