Evaluate the integral. \int \frac{(1+e^{x})^{2}}{e^{x}}dx

korporasidn

korporasidn

Answered question

2021-11-21

Evaluate the integral.
(1+ex)2exdx

Answer & Explanation

Tionant

Tionant

Beginner2021-11-22Added 17 answers

Step 1
The given integral is (1+ex)2exdx.
Step 2
Evaluate the given integral as shown below:
(1+ex)2exdx=1+2ex+e2xexdx
=1ex+2+exdx
=1exdx+2dx+exdx
=1ex+2x+ex+C
Ancessitere

Ancessitere

Beginner2021-11-23Added 17 answers

Step 1: Expand.
1+2ex+e2xexdx
Step 2: Split fraction.
1ex+2exex+e2xexdx
Step 3: Use Sum Rule: f(x)+g(x)dx=f(x)dx+g(x)dx.
1exdx+2dx+exdx
Step 4: Use Integration by Substitution on 1exdx.
Let u=1ex,du=1exdx, then dx=eln1udu
Step 5: Using u and du above, rewrite 1exdx.
u×eln1udu
Step 6: Use this rule: adx=ax+C.
-u
Step 7: Substitute u=1ex back into the original integral.
1ex
Step 8: Rewrite the integral with the completed substitution.
1ex+2dx+exdx
Step 9: Use this rule: adx=ax+C.
1ex+2x+exdx
Step 10: The integral of ex is ex.
1ex+2x+ex
Step 11: Add constant.
1ex+2x+ex+C

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