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

Question

24345 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-15T04:12:15

$sin^6(theta)/6+c$

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

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 $u=sin(theta)$

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

Answer 2 · 2017-11-15T04:13:49

$sin^6(theta)/6 + C$

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 $sin(theta)$ :

$sin^6(theta)/6 + C$