1. Power Rule for Integration (Add one to the exponent, divide by the new exponent)

See if you can simplify the integrand

2. Integration by Substitution

Look for a composite function. Try setting the inside function equal to u or the denominator equal to u. (Don’t forget, you can solve for x in terms of u, if you have an extra x inside the integral)

3. Log Rule for Integration

You need to have a quotient of two functions. Look to see if something close to the derivative of the denominator is in the numerator.

4. Integration Rules for Exponential Functions

If you have e^u inside the integral. You need the derivative of the exponent inside the integral. Once it’s there, it goes away when you integrate and you’re only left with e^u.

5. Long Division

Look for a quotient of two functions. If the degree of the numerator is greater than or equal to the degree of the denominator, try using long division.

6. Completing the Square

If you have a quadratic function in the denominator and you think it might be inverse trig.

7. Integrals of Inverse Trig Functions

Look for a square root in the denominator (it might be arcsin or arcsec). If you have a sum of a function and a constant in the denominator it might be arctan.

8. Integration by Parts

Look for a product of two functions.

If you have an integrand with a single term and you have no idea how to integrate it, try using integration by parts. Let that term be u so that you differentiate it and avoid…