Find the anti-derivative (using substitution)?

Question

#int x^3(x^2+1)^99#
Find the antiderivative of this using the substitution method.
I do know (and understand) that u = #x^2+1# and therefore
#du =##2x# ( #(du)/2=x#). However, I don't understand that even if there's an #x^3#, you just plug it in normally, instead of making #u^3# when you put it in.

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Answer 1 · 2018-03-10T07:57:18

#intx^3(x^2+1)^99dx=1/20200(10x-1)(10x+1)(x^2+1)^100+C#

Let

#I=intx^3(x^2+1)^99dx#

Apply the substitution #u=x^2+1#:

#I=int(u-1)^(3/2)* u^99*(du)/(2sqrt(u-1))#

Simplify:

#I=1/2int(u^100-u^99)du#

Integrate directly:

#I=1/2(1/101u^101-1/100u^100)+C#

Reverse the substitution and simplify:

#I=1/20200(10x-1)(10x+1)(x^2+1)^100+C#