Mixed Integration Practice

NAU STUDENTS:
This exercise gives you new problems to practice—ones requiring renaming before integrating.
Your in-class quiz will cover all the problems from the first integration quiz, together with these new problems.

In this exercise, all functions are assumed to have the required properties for a particular situation.
For example, any function in a denominator is assumed to be nonzero, any function inside a logarithm is assumed to be positive, and so on.

SOMETIMES, YOU JUST NEED TO RENAME..

Sometimes, the function being integrated needs to be re-named, using basic algebra skills.
Perhaps you need to write a radical with a rational exponent.
Perhaps you need to use a distributive law.
Perhaps you need to FOIL something out.
Perhaps you need to use the fact that $\frac{A+B}C = \frac{A}{C} + \frac{B}{C}\,.$

EXAMPLE:

$\displaystyle\int\frac{1}{\root 3\of x}\ dx = \int x^{-1/3}\ dx = \frac{x^{-\frac 13 + 1}}{-\frac13 + 1} + C = \frac{x^{2/3}}{2/3} + C = \frac 32 \root 3\of{x^2} + C $

If a problem starts off in radical form, then you should give your answer in radical form.
Recall that dividing by a fraction is the same as multiplying by its reciprocal: e.g., dividing by $\,\frac 23\,$ is the same as multiplying by $\,\frac 32\,.$
Also recall that for $\,n\ne -1\,,$ $\,\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\,.$

EXAMPLE:

$\displaystyle\int \frac{x^3-1}{x}\ dx = \int \bigl(\frac{x^3}x - \frac 1x\bigr)\ dx = \int( x^2 - \frac 1x)\ dx = \frac{x^3}3 - \ln |x| + C $

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Master the ideas from this section
by practicing the exercise at the bottom of this page.