- Topics At a Glance
- Indefinite Integrals Introduction
- Integration by Substitution: Indefinite Integrals
- Legrange (Prime) Notation
- Leibniz (Fraction) Notation
- Integration by Substitution: Definite Integrals
- Integration by Parts: Indefinite Integrals
- Some Tricks
- Integration by Parts: Definite Integrals
- Integration by Partial Fractions
- Integrating Definite Integrals
- Choosing an Integration Method
- Integration by Substitution
- Integration by Parts
- Integration by Partial Fractions
- Thinking Backwards
- Improper Integrals
- Badly Behaved Limits
- Badly Behaved Functions
- Badly Behaved Everything
- Comparing Improper Integrals
- The
*p*-Test - Finite and Infinite Areas
- Comparison with Formulas
**In the Real World**- I Like Abstract Stuff; Why Should I Care?
**How to Solve a Math Problem**

There are three steps to solving a math problem.

1. Figure out what the problem is asking.

2. Solve the problem.

3. Check the answer.

Find

Answer.

1. Figure out what the problem is asking.

The problem is asking us to find the family of all functions with derivative *x*^{3 }sin *x*. That means we need to use some of our integration techniques.

2. Solve the problem.

We can't use substitution, we can't use partial fractions, and we certainly can't just think backwards from this. That leaves integration by parts.

We want *u'* to be simpler than *u*, so take

*u* = * x*^{3}

*v'* = sin *x*.

Then

*u'* = 3*x*^{2}

*v* = -cos *x*

Apply the magic formula:

To find

we need to use integration by parts again. We still want *u'* to be simpler than *u*, so take

*u* = *x*^{2}

*v'* = cos *x*

*u' *= 2*x*

*v* = sin *x*

Apply the magic formula:

To find

we need to use integration by parts yet again. Take

*u* =* x*

*v'* = sin *x*

*u'* = 1

*v* = -cos *x*

Then apply the formula again:

Now we can go up a step and finish our second integration by parts.

Finally we go back up to the first integration by parts:

Our final answer is

3. Check the answer.

We check the answer by taking its derivative. If we did everything right, we should get *x*^{3 }sin *x*.