How do you integrate $int e^xe^x$ using substitution?

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Answer 1 · 2016-11-20T14:40:13

$inte^xe^xdx$

Method 1 - Immediate Substitution

Make the substitution $u=e^x$ , which implies that $du=e^xdx$ , so

$inte^xe^xdx=int(e^x)(e^xdx)=intudu=u^2/2=(e^x)^2/2=e^(2x)/2+C$

Method 2 - Simplification, then Substitution

Use the rule $a^b(a^c)=a^(b+c)$ to rewrite the integral as

$inte^xe^xdx=inte^(2x)dx$

Now substitute $u=2x$ so $du=2dx$ :

$inte^(2x)dx=1/2int(e^(2x))(2dx)=1/2inte^udu$

Since $inte^udu=e^u$ :

$1/2inte^udu=1/2e^u=e^u/2=e^(2x)/2+C$

How do you integrate int e^xe^xint e^xe^x using substitution?