Substitution with Definite Integrals

Question

If f is continuous and 9 3

∫ f(x) dx = -10, find ∫ x f(x^2)dx

0 0

Answer 1

Let's define u = x². Then du = 2x dx ⇒(1 / 2) du = x dx.

Before we substitute x² with u, let's find the new limit based on u.

Since u = x², when x = 0, u = 0, and when x = 3, u = 9.

When we substitute x² with u in the integral, we need to use the new limits.

Then we have ∫₀⁹ f(u) x dx = (1/2) ∫₀⁹ f(u) du. (since (1/2) du = x dx)

Since we know that ∫₀⁹ f(x) dx = -10, we can conlcude that 1/2 ∫₀⁹ f(u) du = -5.

Answer 2

Start with the question at hand: integral from 0 to 3 of x*f(x^2) dx

u = x^2

du/dx = 2x

dx=du/2x

Upper bound x = 3 gives u = (3)^2 = 9

Lower bound x = 0 gives u = (0)^2 = 0

Replace the integral with new u expressions/values:

integral from 0 to 9 of x*f(u)*du/2x (the x cancels)

1/2 times integral from 0 to 9 of f(u)du

1/2* (-10) = -5