Two integration probs: int e^x(1-e^x)(1+e^x)^10 dx and....

[kagan]

integrate \(\displaystyle \int e^{x}(1-e^{x})%20(1+e^{x})^{10}dx\)

and

integrate \(\displaystyle \int (27e^{9x} + e^{12x})^{1/3}dx\)

please help.

 

[galactus]

For the first one, you could let \(\displaystyle u=1+e^{x}, \;\ du=e^{x}, \;\ u-1=e^{x}\)

Make the subs and get:

\(\displaystyle \int{(2-u)u^{10}}du\)

Now, integrate.

 

[soroban]

Hello, kagan!

\(\displaystyle 2)\;\;\int \left(27e^{9x} + e^{12x}\right)^{\frac{1}{3}}\,dx\)

\(\displaystyle \text{We have: }\;\left[e^{9x}\left(27 + e^{3x}\right)\right]^{\frac{1}{3}} \;=\;\left(e^{9x}\right)^{\frac{1}{3}}\left(27+e^{3x}\right)^{\frac{1}{3}}\)

\(\displaystyle \text{The integral becomes: }\;\int e^{3x}\left(27 + e^{3x}\right)^{\frac{1}{3}}\,dx\)

\(\displaystyle \text{Let }\,u \;=\;27 + e^{3x} \quad\hdots\quad Got\:it?\)