1. Integration Techniques
By: Zoe Horvat, Ally Horwitz, & Noelle Stam

2. Definite vs Indefinite Integrals
*** always remember your cookies (+c) for indefinite integrals!!!
ALLY
Definition of an integral: antiderivative,
Definite integrals have specific upper and lower limits, while indefinite do not
**Always remember your cookies (+c)

3. Basic Antiderivatives
Integration-
Basic Antiderivatives
*** always remember your cookies (+c) for indefinite integrals!!!
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You can take out k from integral
If you have two functions being added together, you can separate them into two different integrals
Integral of du = u + C, general form
Integration of exponential functions: (u)^add 1 to n, then add 1 to n on the bottom, + C
Integral of 1/u(du) = ln|u| + C

4. Integration- U-substitution
Steps:
Choose a value to be U
Take the derivative of U= dU
Sub U and dU back into the original integral
Integrate
Sub U back into antiderivative
Add c (indefinite)
Integration-
U-substitution
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U substitution is a common integration technique you are basically taking/doing the inverse of the chain rule
Explain steps
Explain example problem
*Dont forget + C

5. Using algebraic manipulation to rewrite the integrand
Example:
∫ x(x+2)^1/2 dx
Choose U (U=x+2)
Take the derivative (dU=dx)
Solve for x from U (x=u-2)
Sub in all values into integral
Integrate
ALLY

6. Using trigonometric definitions and properties of exponents and logarithms to rewrite solutions
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Integrals to remember

7. Using geometric interpretations of the definite integral
You will typically use this in free responses and in some multiple choice
ZOE
Area under the curve
Has parts above and below the x-axis
Area above - area below = integral f(x)

8. Integrating by Parts Product rule for integration
Formula: Ultra Violet VooDoo
How to pick u: ILATE
Inverse trig
Log
Algebraic
Trigonometric
Exponential
*** ∫ lnxdx = xlnx-x+C
ZOE
U usually differentiates to zero
Dv is easy to integrate
Basically inverse product rule
Chart of how to pick which value to be u (ILATE)
Important to remember the integral of lnxdx: xlnx-x+c

9. Tabular Method ∫ f(x)g(x)dx F(x) differentiates to 0 eventually
G(x)dx integrates repeatedly
Steps:
Make a table of derivatives of u and integrals of dx
Alternate signs of u
Link terms diagonally
Combine terms diagonally
ZOE
Really fast and easy method, but can only be used when u=x and differentiates to 0
Steps: make a table, u and dx. Derive u and integrate dx until u differentiates to 0
Answer to example: -x^4cos(x) + 4x^3sin(x) + 12x^2cos(x) - 24x(sinx) - 24cos(x)
*Pay attention to signs
*You don’t link first dv or 0 to anything

10. Partial Fractions
We use the method of partial fractions to evaluate certain types of integrals that contain rational fractions
Make sure the degree of the numerator is smaller than the denominator, if not, then long division must be used
Distinct linear factors
ZOE
*In order to use partial fraction method, degree of numerator must be smaller than the denominator
Definition: taking a rational expression and decomposing it into simpler rational expressions
Steps:
Factor denominator as much as possible to get partial fraction decomposition form
Set numerators equal (cross multiply)
Plug in the two values of x to get A and B
Substitute A and B back into partial fraction form
Integrate

11. Improper Integrals An integral is improper if:
Its integrand becomes infinite at one or more points in the interval of convergence
One or both of the limits of integration is infinite
We integrate an improper integral by evaluating the integral for a constant (a) and then taking the limit of the resulting integral as a approaches the limit in question
If the limit exists then the integral converges and the value of the integral is the limit
If the limit doesn’t exist or is positive/negative infinity then the integral diverges and the integral cannot be evaluated there
Improper Integrals

12. Improper Integrals Rules 1 & 2
Rule 1: if f(x) is continuous on [a, ∞)
Rule 2: if f(x) is continuous on (-∞, b]

13. Improper Integrals Rule 3
Rule 3: f(x) is continuous on (-∞, ∞)

14. Improper Integrals Rules 4 & 5
Rule 4: f(x) is continuous on [a,b) and discontinuous at x=b
Rule 5: f(x) is continuous on [a,b) and discontinuous at x=a

15. Improper Integrals Rule 6
f(x) is discontinuous at x=c where a<c<b
Then
***if one integral diverges then the entire integral diverges