Find each indefinite integral and check the result by differentiating.

Question

1) ∫(3x+5)²(3)dx

2) ∫(x²-1)⁵(2x)dx

thank you in advance

Answer 1

Hi Saeed,

These are introductory examples of integration by substitution. Try looking at the integrals to see if you can find both a function f(x) and its derivative f'(x) present in the integrand. If so, making the substitution u = f(x) and du = f'(x)dx may prove fruitful.

For example, suppose you were trying to find ∫(x⁶ + 1)⁴(6x⁵)dx. Since 6x⁵ is the derivative of x⁶ + 1, make the substitution u = x⁶ + 1, du = 6x⁵dx to get ∫u⁴du, which integrates easily to u⁵/5 + C by the Power Rule. Substituting back to get the answer in terms of x gives the result as F(x) = (x⁶ + 1)⁵/5 + C, and the derivative is easily checked to be F'(x) = (x⁶ + 1)⁴(6x⁵) using the Chain Rule.

Answer 2

part 1 Use substitution rule

let u = 3x + 5

du = 3*dx

dx = (du/3)

(1/3) ∫ u² du = (1/3)*(u³/3) + constant = (1/3)* ((3x+5)³/3) + constant. Use the same method for part 2