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48 Cards in this Set
- Front
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Integration by Parts Formula |
∫ u dv = uv - ∫ v du |
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Integration by Parts, Definite Integral Formula |
∫(a to b) u dv = uv|(a to b) - ∫(a to b) v du |
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For ∫ sin^n(x)cos^m(x) dx and n is odd we: |
take a sine out and convert the rest to cosines using sin^2(x) = 1 - cos^2(x), then use the substitution u = cos(x) |
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For ∫ sin^n(x)cos^m(x) dx and m is odd we: |
take a cosine out and convert the rest to sines using cos^2(x) = 1 - sin^2(x), then use the substitution u = sin(x) |
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For ∫ sin^n(x)cos^m(x) dx and n and m are both odd we: |
take out either a cosine or a sine |
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For ∫ sin^n(x)cos^m(x) dx and n and m are both even we: |
use double angle and/or half angle formulas to reduce the integral into a form that can be integrated |
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For ∫ tan^n(x)sec^m(x) dx and n is odd we: |
take a tangent out and a secant out and convert the rest to secants using tan^2(x) = sec^2(x) - 1, then use the substitution u = sec(x) |
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For ∫ tan^n(x)sec^m(x) dx and m is even we: |
take 2 secants out and convert the rest to tangents using sec^2(x) = 1 + tan^2(x), then use the substitution u = tan(x) |
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For ∫ tan^n(x)sec^m(x) dx and n is odd and m is even we: |
we take out either a tan(x)sec(x) or sec^2(x) |
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For ∫ tan^n(x)sec^m(x) dx and n is even and m is odd we: |
deal with each integral sepreatley |
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If integral contains √ (a^2 - x^2), what is the substitution, identity, and quadrant we use? |
u = a sin(ϴ) cosϴ = √ (1 - sin^2(ϴ)) In quadrant 1 and 4 |
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If integral contains √ (a^2 + x^2), what is the substitution, identity, and quadrant we use? |
u = a tan(ϴ) secϴ = √ (1 + tan^2(ϴ)) In quadrant 1 and 4 |
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If integral contains √ (x^2 - a^2), what is the substitution, identity, and quadrant we use? |
u = a sec(ϴ) tanϴ = √ (sec^2(ϴ) - 1) In quadrant 1 and 3 |
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Partial Fractions: ∫ P(x)/Q(x) If P(x) has a smaller degree than Q(x) then we: |
factor the denominator and find the partial fraction decomposition |
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Partial Fractions: ∫ P(x)/Q(x) If P(x) has a larger degree than Q(x) then we: |
use long division to make the degree of P(x) smaller or equal to the degree of Q(x) and then factor the denominator and find the partial fraction decomposition |
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if Q(x) is: ax+b then the PFD is |
A / ax+b |
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if Q(x) is: (ax+b)^n then the PFD is |
A / ax+b + B / (ax+b)^2 + ... + Z / (ax+b)^n |
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if Q(x) is: ax^2+bx+c then the PFD is |
(Ax + B) / (ax^2+bx+c) |
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if Q(x) is: (ax^2+bx+c)^n then the PFD is |
(Ax + B)/(ax^2+bx+c) +...+ (Cx + D)/(ax^2+bx+c)^n |
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convergent integral |
limit exists and is a finite number |
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divergent integral |
limit either doesn’t exist or is (plus or minus) infinity. |
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d/dx (sinx) |
cosx |
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d/dx (cosx) |
-sinx |
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d/dx (tanx) |
sec^2(x) |
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d/dx (secx) |
secxtanx |
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d/dx (cscx) |
-cscxcotx |
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d/dx (cotx) |
-csc^2(x) |
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d/dx (arcsinx) |
1/√ (1-x^2) |
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d/dx (arccosx) |
-1/√ (1-x^2) |
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d/dx (arctanx) |
1/(1+x^2) |
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d/dx (arccotx) |
-1/(1+x^2) |
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d/dx (arcsecx) |
1/(|x|√(x^2 - 1)) |
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d/dx (arccscx) |
-1/(|x|√(x^2 - 1)) |
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d/dx (a^x) |
a^x ln(a) |
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d/dx (ln(x)) |
1/|x| |
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d/dx (log_a (x) |
1/(xlna) |
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∫ 1/ (ax+b) dx |
(1/a) ln|ax+b| + C |
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∫ (1/x) dx |
ln|x| + C |
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∫ cos x dx |
sin x + C |
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∫ sin x dx |
-cos x + C |
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∫ sec^2 x dx |
tan x +C |
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∫ secxtanx dx |
sec x + C |
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∫ cscxcotx dx |
-cscx + C |
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∫csc^2 x dx |
-cotx + C |
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∫ tan x dx |
ln|secx| + C |
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∫cot x dx |
ln|sin x| + C |
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∫ sec x dx |
ln|sec x +tan x| + C |
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∫ csc x dx |
ln|cscx-cotx| + C |