Question

In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{-1}^{3} \frac{x d x}{x^{2}+1}$$

Short answer

The value of the integral is \(\frac{1}{2} \ln 5\).

Step-by-step solution

Make the substitution

Substitute \(u = x^{2}+1\). Then compute the differential \(du\), which is given by \(du = 2x \, dx\). As we only have \(x \, dx\) in our integral, we express \(x \, dx\) as \(x \, dx = \frac{1}{2} \, du\).

Change the limits of integration

Since we are performing a definite integration, we must alter our limits accordingly with the substitutions. Substituting the initial limit \(x = -1\) into \(u = x^{2}+1\) gives \(u(-1) = 2\). Using the final limit \(x = 3\) gives \(u(3) = 10\). Thus the integration bounds have been changed from [-1, 3] to [2, 10].

Substitute into the integral

Under the substitution, the integral becomes \(\int_{2}^{10} \frac{du}{2u}\). This simplifies as \(\frac{1}{2} \int_{2}^{10} \frac{du}{u}\).

Evaluate the integral

The integral of \(\frac{1}{u}\) with respect to \(u\) is \( \ln |u|\). Hence, the antiderivative of the function in the integral is \(\frac{1}{2} \ln |u|\). Evaluating this from 2 to 10 we have: \(\frac{1}{2}[\ln|10| - \ln|2|] = \frac{1}{2} \ln 5\).

Final answer

The value of the integral is \(\frac{1}{2} \ln 5\).