integration by parts with large trig functions

May 2010
2
0
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
Jun 2008
16,217
6,765
North Texas
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 \(\displaystyle u = \cos{x}\)

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

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

\(\displaystyle \int_0^1 u^{18} - 2u^{20} + u^{22} \, du\)
 
  • Like
Reactions: gogogo and AllanCuz
Similar Math Discussions Math Forum Date
Calculus
Calculus
Calculus
Calculus