Math 106: Review for Exam II
INTEGRATION TIPS
•Substitution: usually let w= an inside function, especially if w0is also present in the integrand
•Parts: Zudv=uv −Zvdu or Zuv0dx =uv −Zu0vdx
How to choose which part is u? Let ube the part that is higher up in the LIATE mnemonic below.
(The mnemonics ILATE and LIPET will work equally well if you have learned one of those instead;
in the latter Ais replaced by P, which stands for “polynomial”.)
Logarithms (such as ln x)
Inverse trig (such as arctan x, arcsinx)
Algebraic (such as x, x2,x
3+4)
Trig (such as sin x, cos 2x)
Exponentials (such as ex,e
3x)
•Rational Functions (one polynomial divided by another): if the degree of the numerator is greater than
or equal to the degree of the numerator, do long division then integrate the result.
Partial Fractions: here’s an illustrative example of the setup.
3x2+11
(x+ 1)(x−3)2(x2+5) =A
x+1+B
x−3+C
(x−3)2+Dx +E
x2+5
Each linear term in the denominator on the left gets a constant above it on the right; the squared
linear factor (x−3) on the left appears twice on the right, once to the second power. Each irreducible
quadratic term on the left gets a linear term (Dx +Ehere) above it on the right.
•Trigonometric Substitutions: some suggested substitutions and useful formulae follow.
Radical Form pa2−x2pa2+x2px2−a2
Substitution x=asin t x =atan t x =asec t
sin2x+ cos2x= 1 tan2x+ 1 = sec2x
sin2x=1
2−cos(2x)
2cos2x=1
2+cos(2x)
2
sin(2x) = 2 sin xcos x
•Powers of Trigonometric Functions: here are some strategies for dealing with these.
Zsinmxcosnxdx Possible Strategy Identity to Use
modd Break off one factor of sin xand substitute u= cos x. sin2x=1−cos2x
nodd Break off one factor of cos xand substitute u= sin x. cos2x=1−sin2x
m, n even Use sin2x+ cos2x= 1 to reduce to only powers of sin xsin2x=1
2−cos(2x)
2
or only powers of cosx, then use table of integrals #39-42 cos2x=1
2+cos(2x)
2
or identities shown to right of this box.
Ztanmxsecnxdx Possible Strategy Identity to Use
modd Break off one factor of sec xtan xand substitute u= sec x. tan2x= sec2x−1
neven Break off one factor of sec2xand substitute u= tan x. sec2x= tan2x+1
meven, nodd Use identity at right to reduce to powers of sec xalone. tan2x= sec2x−1
Then use table of integrals #51.
