Question

`int x^(3)tan^(-1)(x^(2))dx`

Answer 1

Answer:

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Step-by-step explanation:

∫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