Definite Integrals Question 17

The function $f$ is continuous and $\int\limits_0^8 f(u) du = 10$ . What is the value of $\int\limits_1^3 xf(x^2-1) dx$ ?

5

10

20

40

Approach

\int\limits_1^3 xf(x^2-1)  dx

Use the following substitutions:

u=x^2-1 \hskip{2em} du=2x  dx \hskip{2em} dx=\frac{1}{2x}du

\int\limits_0^8 xf(u)\cdot \frac{1}{2x}  du =\frac{1}{2}\int\limits_0^8 f(u)  du

= \frac{1}{2}(10) =\boxed{5}