How to solve this integration? Integrate x^3 tan^-1(x) dx

Question

Need the integration of 
x^3 tan^-1(x) dx for my assignment.
Please someone help!!!!

Answer 1

∫x³tan^-1(x)dx

 

Use integration by parts.

 

 

Let       u = tan^-1x                      dv = x³dx

         du = 1 / (1 + x²)dx             v = (1/4)x⁴

 

Then the integral is

 

(1/4)x⁴tan^-1x - (1/4)∫[x⁴ / (1 + x²)]dx

 

 

Then, you take the integral of the sub-integral.

 

∫x⁴ / (1 + x²)dx

 

using trig substitution.

 

x = tanθ                        x² = tan²θ

dx = sec²θdθ                 x⁴ = tan⁴θ

 

∫[(tan⁴θ) / (1 + tan²θ)] sec²θdθ

 

Since              sec²θ = 1 + tan²θ

 

∫tan⁴θdθ

 

 

Use your trig identities to find this integral.  Then substitute back into the variable x and put the integral all together.