Managerial Calculus (Evaluate the following integrals)

Evaluate the following integrals:

1)
∫x^1/2(x-1)(x-9)dx

2)
⌠2
⎮Inx/x dx
⌡1

3)
⌠1
⎮x√(x^2 + 1)dx
⌡0

4)
⌠1
⎮e^5x / e^3x dx
⌡0

1) √(x) (x - 1) (x - 9)
= x^(1/2) (x² - 9x - x + 9)
= x^(1/2) (x² - 10x + 9)
= x^(2+1/2) - 10x^(1+1/2) + 9x^(1/2)
= x^(5/2) - 10x^(3/2) + 9x^(1/2)
Integrate that, term by term, using the power rule

2) let u = ln(x)
du = 1/x dx
Substitute and integrate the resulting integral with the power rule (∫ u du).

3) let u = x² + 1
du = 2x dx
Substitute and integrate using the power rule (∫ u^(1/2) du)

4) e^(5x) / e^(3x) = e^(5x - 3x) = e^(2x)
let u = 2x
du = 2 dx
Substitute and integrate using the definition of e (∫ e^u du = e^u)