Evaluate the following integrals. ∫exe2x+2ex+17dx

bobbie71G

bobbie71G

Answered question

2021-10-20

Evaluate the following integrals.
exe2x+2ex+17dx

Answer & Explanation

jlo2niT

jlo2niT

Skilled2021-10-21Added 96 answers

Given that:
The integral is exe2x+2ex+17dx
By using the formula,
1x2+a2dx=1aarctan(xa)+C
By using substitution,
Substitute v=ex, then dv=exdx
Then,
exe2x+2ex+17dx=1v2+2v+17dv
=1v2+2v+11+17dv
=1(v2+2v+1)+16dv
=1(v+1)2+16dv
Substitute u=v+1, du=dv
Then, =1u2+16du
=1u2+42du
=14arctan(u4)
Substitute back u=v+1 and v=ex, then u=ex+1
exe2x+2ex+17dx=14arctan(u4)
=14arctan(ex+14)+C
Where C is the integration constant.
Therefore,
exe2x+2ex+17dx=14arctan(ex+14)+C, where c is the integration constant.

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