Evaluate the integral. \int \frac{e^{x}+e^{3x}}{e^{2x}}dx

IMLOG10ct

IMLOG10ct

Answered question

2021-11-21

Evaluate the integral.
ex+e3xe2xdx

Answer & Explanation

George Morin

George Morin

Beginner2021-11-22Added 13 answers

We know that,
eaxdx=eaxa
Evaluating the integral.
ex+e3xe2xdx=(exe2x+e3xe2x)dx
=(1ex+ex)dx
=1exdx+exdx
=exdx+exdx
=ex1+ex+C, where C is integration constant
=ex+ex+C
=exex+C
Therefore,
ex+e3xe2xdx=exex+C
Theirl1972

Theirl1972

Beginner2021-11-23Added 22 answers

Step 1: Simplify ex+e3xe2x  1ex+ex.
1ex+exdx
Step 2: Use Sum Rule: f(x)+g(x)dx=f(x)dx+g(x)dx.
1exdx+exdx
Step 3: Use Integration by Substitution on 1exdx.
Let u=1ex,du=1exdx, then dx=eln1udu
Step 4: Using u and du above, rewrite 1exdx.
u×eln1udu
Step 5: Use this rule: adx=ax+C.
-u
Step 6: Substitute u=1ex back into the original integral.
1ex
Step 7: Rewrite the integral with the completed substitution.
1ex+exdx
Step 8: The integral of ex is ex.
ex1ex
Step 9: Add constant.
ex1ex+C

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