# integration by parts with large trig functions

#### gogogo

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.....

#### skeeter

MHF Helper
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 $$\displaystyle u = \cos{x}$$

$$\displaystyle du = -\sin{x} dx$$

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

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

gogogo and AllanCuz

#### gogogo

wooooow! thanks!!!

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