The following integral requires a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate the following integral. x^4 + 7/x^3 + 2x dx Find the partial fraction decomposition of the integrand. x^4 + 7/x^3 + 2x dx

Division yields

Now for partial fractions: you're looking for constants a, b, and c such that

which gives a + b = 2, c = 0, and 2a = -7, so that a = -7/2 and b = 11/2. Then

Now, in the integral we get

The first two terms are trivial to integrate. For the third, substitute y = x ² + 2 and dy = 2x dx to get