my problem is to integrate (sin^5(x))(cos^18(x)). Lower limit is 0 and upper limit is pi/2. I tried graphing this and using calculators because my teacher didn't explain that far into using integration by parts.....

straight substitution will work ...

\(\displaystyle -\int_0^{\frac{\pi}{2}} \sin^4{x} \cdot \cos^{18}{x}(-\sin{x}) \, dx\)

\(\displaystyle -\int_0^{\frac{\pi}{2}} (1-\cos^2{x})^2 \cdot \cos^{18}{x}(-\sin{x}) \,

dx\)

let \(\displaystyle u = \cos{x}\)

\(\displaystyle du = -\sin{x} dx\)

\(\displaystyle -\int_1^{0} (1-u^2)^2 \cdot u^{18} \, du\)

\(\displaystyle \int_0^1 u^{18} - 2u^{20} + u^{22} \, du\)