Trigonometric Substitution

Question

Using the substitution x=2 tan (θ), evaluate the indefinite integral of

(4+x^2)^2

Thank you in advance.

Answer 1

I = ∫(4+x²)²dx

x=2tan(θ), dx = 2sec²(θ)dθ, tan(θ)=x/2

∫(2 (4²) (1+tan²(θ))²sec²(θ)dθ

Let u = tan(θ), du = sec²(θ)dθ

32∫(1+u²)²du = 32∫u⁴+2u²+1du = 32(u⁵/5+u³/3+u) = 32(tan⁵(θ)/32+2tan³(θ)/8+tan(θ)/2)

= x⁵ + 8x³/3 + 16x + C