Evaluate the integral by making the given substitution. (Use C for the constant of integration.)?

enter image source here

2 Answers
Nov 15, 2017

sin^6(theta)/6+c

Explanation:

intsin^5(theta)cos(theta)d(theta)

with u=sin(theta)

then the derivative is
du=cos(theta)d(theta)

d(theta)=1/cos(theta)
now in the integral

int(u)^5cos(theta)1/cos(theta)d(theta)

the cos is goes

int(u)^5d(theta)

integrating

int(y)^ndy=y^(n+1)/n

int(u)^5d(theta)=u^6/6+c

but u=sin(theta)

intsin^5(theta)cos(theta)d(theta)=sin^6(theta)/6+c

Nov 15, 2017

sin^6(theta)/6 + C

Explanation:

Using U-Substitution:
u = sin(theta)
du = cos(theta)

Change the variables in your integral:
int sin^5(theta)cos(theta)d theta

int u^5du

Integrate:
int u^5du=

=u^6/6 + C

Replace u with sin(theta):

sin^6(theta)/6 + C