Integration Strategy
Calculus II
Josh Engwer
TTU
05 March 2014
Josh Engwer (TTU)
Integration Strategy
05 March 2014
1/5
Relevant Trig Identities
Memorize these:
OPP
ADJ
OPP
HYP , cos θ := HYP , tan θ := ADJ
HYP
ADJ
csc θ := OPP
, sec θ := HYP
ADJ , cot θ := OPP
csc θ = sin1 θ , sec θ = cos1 θ , cot θ = tan1 θ
sin θ
cos θ
tan θ = cos
θ , cot θ = sin θ
2
2
2
2
sin θ :=
sin θ + cos θ = 1, tan θ + 1 = sec θ, 1 + cot2 θ = csc2 θ
sin (−θ) = − sin θ, cos (−θ) = cos θ
sin (A + B) = sin A cos B + cos A sin B
cos (A + B) = cos A cos B − sin A sin B
sin (A − B) = sin A cos B − cos A sin B
cos (A − B) = cos A cos B + sin A sin B
sin (2θ) = 2 sin θ cos θ
cos2 θ =
1+cos(2θ)
, sin2
2
Josh Engwer (TTU)
θ=
1−cos(2θ)
2
Integration Strategy
05 March 2014
2/5
Basic Integral Rules
Memorize these:
Z
xn dx
Z
ex dx
Z
sin x dx
Z
sec2 x dx
Z
sec x tan x dx
Z
tan x dx
Z
sec x dx
Josh Engwer (TTU)
xn+1
=
+C
n+1
Z
= ex + C
= − cos x + C
= tan x + C
= sec x + C
= ln | sec x| + C
= ln | sec x + tan x| + C
1
dx
Z x
ax dx
Z
cos x dx
Z
csc2 x dx
Z
csc x cot x dx
Z
cot x dx
Z
csc x dx
Integration Strategy
= ln |x| + C
=
ax
+C
ln a
= sin x + C
= − cot x + C
= − csc x + C
= ln | sin x| + C
= − ln | csc x + cot x|
05 March 2014
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Integration Toolbox
Algebraic Simplification
Factoring
Binomial Theorem/Pascal’s Triangle
Rationalizing the Numerator (RN)
Rationalizing the Denominator (RD)
Clever insertion of one (CI-1)
Clever insertion of zero (CI-0)
Split Fraction (SF)
Trig Identities (TRG)
Reference Triangles
Basic Integral Rules
Change of Variables (CV)
Integration by Parts (IBP)
Partial Fraction Decomposition (PFD)
Completing the Square (CS)
Forms involving powers
of √
trig functions√
√
Forms involving a2 − u2 , a2 + u2 , or u2 − a2
Josh Engwer (TTU)
Integration Strategy
05 March 2014
4/5
Fin
Fin.
Josh Engwer (TTU)
Integration Strategy
05 March 2014
5/5