By substituting u = f(x), we were able to rewrite the original integral into a simpler form with respect to u. The new integral is ∫ fsin(u) du, with limits of integration from 4 to 9.
Rewriting the Integral using u-substitution
Problem:
Rewrite the integral
∫ ff'(x)sin (f(x)) dx
as an integral with respect to u, given that f(1) = 4 and f(3) = 9.
Solution:
Let u = f(x). Then du = f'(x) dx. Substituting these into the integral, we get:
∫ ff'(x)sin (f(x)) dx = ∫ fsin(u) du
Finding the limits of integration:
We need to find the limits of integration in terms of u. We know that f(1) = 4 and f(3) = 9, so:
When x = 1, u = f(1) = 4.
When x = 3, u = f(3) = 9.
Therefore, the new limits of integration are 4 and 9.
Final integral:
The integral after u-substitution is:
∫ fsin(u) du, with limits of integration from 4 to 9.