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

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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#