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Chapter 5: Problem 5

When using a change of variables \(u=g(x)\) to evaluate the definite integral \(\int_{a}^{b} f(g(x)) g^{\prime}(x) d x,\) how are the limits of integration transformed?

Short Answer

Expert verified
Answer: The limits of integration after the change of variables will be \(u_a = g(a)\) and \(u_b = g(b)\).

Step by step solution

01

Evaluate the function at the limits of integration

To find the new limits of integration, transform the old limits \(a\) and \(b\) with the substitution \(u=g(x)\). Compute the values of \(u\) at \(x=a\) and \(x=b\): $$ u_a = g(a) \\ u_b = g(b) $$
02

Replace the limits of integration

Now replace the original limits of integration \(a\) and \(b\) with the new limits in terms of the variable \(u\). So, the transformed definite integral will be, $$ \int_{u_a}^{u_b} f(u) \frac{d u}{d x} d u $$
03

Simplify the integral

Notice that the term \(\frac{d u}{d x}\) is actually the derivative of the function \(g(x)\) with respect to \(x\). From the change of variables, we have \(du = g'(x) dx\). So we can rewrite the transformed definite integral in a more simplified manner: $$ \int_{u_a}^{u_b} f(u) d u $$ The limits of integration have been transformed according to the function \(g(x)\), and the definite integral can now be evaluated with respect to the new variable \(u\).

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