Integration Techniques

Sin/Cos (odd powers)

1. Factor out one term (which will be du) 2. Take the integral of du to find what u should equal 3. Turn the remaining (even powers) into the other trig function (which will be u) 4. Replace remaining piece into u using the following identity sin2 x + cos2 x = 1 *If both or odd; either work*

Case 1: Linear Factors not repeated

1. Factor the denomenator into factored form and each piece will be the denomenator of the new form 2. Now left with original top/factored bottom=A/first piece + B/second piece + C/third piece +... 3. Multiply both sides by the factored bottom and left with original top= A(side B)(side C) + B(side A)(side C) + C(side A)(side B) +... 4. Solve for A,B, and C using one of two methods Method 1: works everytime: A. Multiply inside parenthesis each piece and distribute thus removes all the parenthesis B. Now introduce new parenthesis by having common factor outside multiplied by 3 or more or less pieces (usually the pieces on outside match the original function) C. Form 3 equations to solve for the three unknowns (three combined factors forms) D. Solve for A, B, and C Method 2- use first unless gets you stuck then switch to method 1 (assumes that the equalities work for all values of x) (this doesnt work for irreduciable because no factor to equal 0) A. Pick different x-values to get the factored forms to equal 0 B. One value pick will make 2 equal 0 because each factor seen twice *Once solve for a piece can plug into the equation* 5. Once A, B, and C are found, plug into the integral of A value over bottom 1 + B value over 2 + C value over bottom 3 +... 6. Solve integral (+c unless endpoints)

Strategies for integration

1. Multiply through, u-sub, simplify 2. If there is trig involved and u-sub doesnt work, change everything to sin.cos or use an identity 3. if poly/poly and u sib doesnt work; partial fractions 4. the product of two unrelated functions (can't combine them)- use integration by parts (or use if dont know how to integrate one of the pieces but know its derivative) 5. Only trig involved- trig substitution (triangles) or identites (if turning into sin/cos/simplification doesn't work 6. Anything with a radical- check if straight up inverse trig, if not try trig sub (triangles) *May need to combine methods*

Sin/Cos (even powers)

1. Pick the identity that closely resembles the problem from 1/2 (1 − cos 2x) = sin2 x 1/2 (1 + cos 2x) = cos2 x 1/2 sin 2x = sin x cos x (endpoints can remain because no variable change) 2. Attempt to solve; may require u-sub, using the identity again, etc.

U-Substitution

1. figure out what u = (it will be something that is plugged into another thing) 2. differentiate u to get du= u' dx 3. tweak equation by multiplying to get a u' if needed 4. if have an extra x then put in terms of u (ex. u-1/u+1) 5. plug in u and du and integrate 6. dont forget to add c unless endpoints are given (can switch back to x or find out what u is given what x is)

A: If Degree on top larger or equal to denomenator

Do long division Usually left with a remandier and that uses u-sub Or break up again with 1 or 2

B: Degree top less

Factor the denomenator as much as possible and left with 4 possible cases which says where to go next

Sin/Cos Coefficents inside greater than 1

For the integrals involving sin x and cos x with coefficients inside the trig more than one, use the given identities: sin(mx) cos(nx) dx; sin A cos B = 1 2 [sin(A − B) + sin(A + B)] sin(mx) sin(nx) dx; sin A sin B = 1 2 [cos(A − B) − cos(A + B)] cos(mx) cos(nx) dx; cos A cos B = 1 2 [cos(A − B) + cos(A + B)] 1. Pick the most similar identity 2. Plug #'s in 3. Solve seperate integrals (split up from +/-) (May need to use U-sub or any other technique)

SecxTanx

Guidelines If sec x has an even power, factor out a sec2 x, change the remaining factors to be in terms of tan x using 1 + tan2 x = sec2 x. The let u = tan x, du = sec2 x dx If tan x has an odd power, factor out a sec x tan x and change the remaining factors to be in terms of sec x using tan2 x = sec2 x − 1. Then let u = sec x, du = sec x tan x dx. Other cases (ex: tan odd no sec or tan even) use identities, integration by parts, rewrite in terms of sin x, cos x, ingenuity! You can also use cot2 x = csc2 x − 1 and csc2 x = 1 + cot2 x May need to manipulate to get a tan or sec into the problem

Case 4: Repeated Quatratic Factors

Identical to linears with repeating Form coresponding numerators and factored denominators with increasing powers Multiply both sides by denomenator Solve for numerators (sames ways as before)

Trig Substitution

Often has a square root and resembles inverse trig Use triangle method 1. Find three pieces of triangle (whole square root, square root of first piece, and square root of second piece) (three options square root a^2-b^2x^2; square root b^2x^2-a^2; and square root a^2+b^2x^2) 2. Place on triangle (if + under square root that is hyp because adding if not hyp is first piece because cant have negative under the square root 3. Once found hyp the simplest remaining piece containing x is opposite the angle 4. Solve for x, dx, and the square root 5. Plug into the integral and solve (don't want anything in terms of x all to be theta) 6. Now things should cancel 7. May need to change endpoints multiple times (x to theta then to new variable (or can plug back a couple steps but this is easier)

Improper Integrals

Only occur with endpoints Used if there is a discontinuity between a and b or if one of the enpoints is infinity Infitite endpoints: integral from a to infinite/neg infinite to b; of function dx 1. have to take the limit as t approaches the value 2. Solve integral as normal then take the limit If get a finite number- convergent If get an infinite number- divergent If neg infinity to infinity- 1. break up into two integrals (ideally 0) and go from neg infinity to 0 and 0 to infinity Discontinuity: break up at the discontinuity and take the limit as t/s approaches c

Case 2: Repeated Linear Terms

Same as Case 1 but for each repeating term with increasing powers in the denomenator until all are counted Numerators are A,B,C,D,E 1. Multiply denomenators 2. Solve for A,B,C,D,E (plug in x-values or distribute and form new paranthesis) 3. Plug in and solve for integral, if a c is involved use +K

Integration Tricks

Sin^4x=(sin^2x)^2 Multiply inside the integral same top and bottom because it =1 Change to terms of sin/cos or use pythagreon identities Trig identities Break up integral into seperate integrals where there are plusses and minuses Try U-sub first everytime

Integration by Parts

Undoes the chain rule (used if have 2 equations multiplied together that don't have anything in common) 1. Figure out what u and dv are going to be (guidelines are to pick a u that you don't know how to integrate/most complicated term; if know how to derive both pick u to be the term that once the derivative is taken, becomes a simpiler value (ex: not trig or exponential)) 2. Find du and v 3. Plug into equation integral of udv=uv-integral of vdu 4. Simplify and solve (may have to use u sub, integration by parts again, or simplification) 5. +c unless endpoints (endpoints go to both the uv and the integral)

Case 3: Irreducible Quadratic Factors

Used when there is an x^2+c not (x+c)^2 1. Factor (if top bigger still do long division) 2. The numerators will be Ax+B, Cx+D... and denomenator each of the factored pieces *Only quadratics have AX=B if linear still in problem those would be just A* 3. Solve for numerators; linears can be done with factors for quadratics plug in linears found and do method 1: distribute remove (), group together same variable terms form new (), and set equal to the matching value from the original numerator

Integration by Partial Fractions

Usually used with fractions with polynominals Idea is to split into two fractions, therefore creating two integrals and solve with u-sub/partial fractions again

Basic Integration Rules

d/dx arcsin 1/sqrt1-x^2 arctan 1/1+x^2 arcsec 1/|x|sqrtx^2-1 co functions all negative integrals of those arcsin u/a+c 1/aarctanu/a+c 1/aarcsec|u|/a+c