Integration by Parts
Trig. Integrals and Trig Sub.
Partial Fraction Decomposition
Improper Integrals
Random
100
What is the Formula for IBP? How do you choose u/dv
intudv=uv-intvdu
"Ultra Violet Super Voodoo"
DETAILS/ILATE
100
What are the Pythagorean, Double Angle, and Power Reduction Ids?
Pythagorean:
sin^2theta+cos^2theta=1
tan^2theta+1=sec^2theta
Double Angle:
sin(2theta)=2sinthetacostheta
cos(2theta)=2cos(2theta)-1
Power Reduction:
sin^2theta=1/2(1-cos(2theta))
cos^2theta=1/2(1+cos(2theta))
100
Decompose the Fraction:
x^6/(x^2-4)
x^4+4x^2+16+16/(x-2)-16/(x+2)
100
What are the 2 types of Improper integrals?
Type I: Infinite Interval
int_a^(oo)f(x)dx=lim_(t to oo)int_a^tf(x)dx
Type II: Discontinuous Integrand
Suppose f is undefined at c in [a,b]
int_a^bf(x)dx=lim_(t to c^-)int_a^tf(x)dx+lim_(t to c^+)int_t^bf(x)dx
100
int_1^2x/(x+1)^2 dx
ln(3/2)-1/6
200
intdt/(2t^2+3t+1)
ln|2t+1|-ln|t+1|+C
200
Does this integral converge?
int_1^oo1/x^(sqrt2)dx
Yes by p-test
sqrt(2)~~1.41>1
200
intxsec(x)tan(x)dx
xsec(x)-ln|sec(x)+tan(x)|+C
300
int(x^2+8x-3)/(x^3+3x^2)dx
3ln|x|+1/x-2ln|x+3|+C
300
int_0^1(x-1)/sqrtxdx
-4/3
300
intdx/(e^xsqrt(1-e^(-x^2)))
-sin^-1(e^-x)+C
400
int(x^3-4x-10)/(x^2-x-6)dx
1/2x^2+x+ln|x-3|+2ln|x+2|+C
400
Use the Comparison Test to determine if the following integral converges:
int_1^oox^3/(x^5+2)dx
converges
x^3/(x^5+2)<=x^3/x^5=1/x^2
and
int_1^oo1/x^2dx
converges by p-test as 2>1
400
int_(-1)^1dx/(x^2-2x)
500
int(3x^3-x^2+6x-4)/((x^2+1)(x^2+2))dx
3/2ln|x^2+1|-3tan^-1(x/sqrt2)+C
500
int_1^oodln(x)/(x^4)dx
Use IBP
=1/9