Similar functions can have very different antiderivatives. For example, even though though looks very similar to (the functions are reciprocals), their solutions are very different. Prove that I'm right by solving both.

Difficulty:

Solution:

The first integral is of the form , where and a = 1. Ignore the x in the numerator for now. Do you recognize that pattern? Its antiderivative, one you apply u-substitution, is . Since , du = 2xdx and . With all of this information, you can rewrite the integral and solve:

The other integral is much easier to solve. Begin by splitting the fraction into two smaller fractions of the same denominator:

Simplify and find the antiderivatives. Your result will be very different from the other integral we just finished:

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Find the antiderivative of .

Difficulty:

Solution:

Since the denominator is factorable, integration by parts is a dead giveaway. Start by factoring the denominator:

Our goal is to write this fraction as a sum of smaller fractions, whose denominators are pieces of this factored denominator:

Multiply both sides of the equation by x(x + 2)(x - 2), the least common denominator:

The only constant term on the left is "+1". The constant term on the right is "-4A". If these equations are to be equal, the constant pieces must be equal. This allows us to find A:

We also know that there is no quadratic (degree 2) term on the left side, but there is one on the right. This cannot be! We need to make sure the one on the right goes away so the sides of the equation match:

This gives us an equation in two variables. We also know that , so that the x terms on both sides of the equation match. If you solve the system of equations for this and the last equation we came up with, you find that . Now that you know the values of the three variables, you can substitute in the values and integrate as appropriate. If you're feelin' extree crazy, you can apply logarithmic properties to get a really compact but bizarre version of the solution: