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

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)

Basic Antiderivatives Integration- Basic Antiderivatives *** always remember your cookies (+c) for indefinite integrals!!! ALLY 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

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 ALLY 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

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

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

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)

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

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

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

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

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

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

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

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