Question

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

Answer 1

`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`