Evaluate the integral using the indicated trigonometric substitution.

Question

∫5x^3(sqrt(9−x^2))dx and x =3sin(θ)

Answer 1

Cannot evaluate the integral without the limits given, and there are none given.

but ∫ 5x³(9-x²)dx can be simplified using

let x = 3 Sin(θ)

dx = 3Cos(θ) dθ

= ∫5(3³Sin³θ(9 - 9Sin²θ)^1/2 [3Cosθ]dθ

simplifying

10935∫Sin³θCos³θ dθ

Using an Integral Calculator gives a messy answer.

−[sin4(x)(2sin2(x)−3)]/12+C