Evaluate the integrals

Question

1) ∫(4x^2-12x+9)^(5/2) dx

2) ∫(csc^2 √x)/(√x) dx

Answer 1

1) The integrand is [(2x - 3)²]^(5/2) = (2x - 3)⁵ (i.e., the exponents 2 and 5/2 multiply)

After factoring the integrand, I'll multiply and divide by 2 as follows:

∫(2x - 3)⁵dx = 1/2 * ∫(2x - 3)⁵2dx. Since 2dx is d/dx of the parenthesized expression, this is equal to

1/2 * 1/6 * (2x - 3)⁶ + C, or (2x - 3)⁶/12 + C

2) Set u = √x. Then du = 1/(2√x) * dx

The integral becomes 1/2 * ∫csc²(u)du = -1/2 * cotan(u) + C = -1/2 * cotan(√x) + C