Exam #2 Review

Question 1

Integration by Parts

Answer 1

Integral of udv = uv - integral of vdu
Id u and dv. LIATE for picking dv (exponents, trig, algebra, inverse, logs)
May have to repeat process more than once
Don’t forget +C for indefinite integrals

Question 2

Trig intrgrals

Answer 2

Uses trig identities to get at a u sub
sin even power, cos odd power: pull off one cos, write remaining cos in terms of sin, then u sub for sin. vice versa.
IF u=cos(x), then du=sin(x)dx
IF u=sin(x), then du=cos(x)dx
IF u=tan(x), then du=sec2(x)dx
IF u=sec(x), then du=sec(x)tan(x)dx
u=trig, then integral of one is u
u=not trig, then integral of one is x

Question 3

Double angle for sin

Answer 3

sin(2x)=2sin(x)cos(x)

Question 4

Integral of ln(x)

Answer 4

xln(x)-x+c

Question 5

Trig sub

Answer 5

square root of a2x2-b2 => sec2(q)-1=tan2(q)
square roof of a2x2+b2 => tan2(q)+1=sec2(q)
square root of b2-a2x2 => 1-sin2(q)=cos2(q)
find x=
find dx=
convert back to x! may need triangle

Question 6

Changing bounds from x to theta

Answer 6

x=0 sub of x=sin(q), so sin(0)=0 q bound = 0

x=1 sub of x=sin(q), so sin(1)=pi/2 q bound = pi/2

Question 7

Partial Fraction Decomposition

Answer 7

integral of polynomial P(x)/polynomial Q(x)
must be proper (degree of P(x) A/(x+3)
-Repeated linear terms (x+3)^2 => A/(x+3) + B/(x+3)^2
-Irreducible quadratic (x^2 +1) => Ax+B/(x^2 +1)
-Repeated irreducible quadratic (x^2 +1)^2 => Ax+B/(x^2 +1) + Cx+D/(x^2 +1)^2

Question 8

Finding PFD numerators

Answer 8

1. multiply through by common denominator
2. choose x values to solve for A, B, C, etc.
3. Substitute in those values to find the last one, if needed
4. DON’T FORGET TO INTEGRATE ONCE DONE FORMING PFD!!!!!