Question

if the change of variables $u = x^{2}+5$ is used to evaluate the definite integral $int_{3}^{9}f(x)dx$, what are the new limits of integration?

Explanation:

Step1: Find lower - limit value

When \(x = 3\), substitute into \(u=x^{2}+5\).
\[u=(3)^{2}+5=9 + 5=14\]

Step2: Find upper - limit value

When \(x = 9\), substitute into \(u=x^{2}+5\).
\[u=(9)^{2}+5=81 + 5=86\]

Answer:

The new lower - limit is 14 and the new upper - limit is 86.