AB Calc Review

Partial Fraction Integration

100

∫(x+5)/(x-3)(x+1)

2ln|x-3|-ln|x+1|+c

200

∫2x+3/(x+2)(x-1)

(⅓) ln|x+2| + (5/3) ln|x-1|+c

300

∫(3x-5)/(x-3)²

3ln|x-3|-(4/x-3)+c

400

∫(3x-1)/(x-2)²(x+1)

(4/9)ln|x-2|-(5/3)/(x-2)-(4/9)ln|x+1|+C

500

∫(3x²+4x+5)/(x+1)(x²+4)

(4/5)ln|x+1|+(11/10)ln|x²+4|+(9/10)arctan(x/2)+c

Integration by Parts Day 1

100

Why use it

To separate an integral you can't otherwise do

200

Formula

∫udv=uv−∫vdu.

300

What would u and v be in the following problem 

∫x^2e^xdx

u = x^2

v = e^x

400

∫xe2xdx

e^2x(x/2-1/4) +C

500

∫x2ln(x)dx

((x^3)/3)ln(x)- (x^9)/3 + C

Integration by Parts Day 2

100

∫x²e^xdx

e^x(x²-2x+2)+C

200

∫x²sin(x)dx

-x²cos(x) + 2xsin(x) + 2cos(x) + C

300

∫x³ln(x)dx

x⁴/4 ln(x) - x⁴/16 + C

400

∫x²e^2xdx

e^2x(x²/2 - x/2 +1/4) + C

500

∫e^xcos(x)dx

e^x/2(sin(x) + cos(x)) + C

Rapid Repeated Integration by Parts

100

∫(e^x)(5x³-x²+4x+3)dx

e^x(5x³-16x²+36x-33)+C

200

∫(x²)(e^-2x)dx

(x²e^-2x/-2)-(xe^-2x/2)-(e^-2x/4)+c

300

∫(x⁶e^4x)dx

(x⁶e^4x/4)-(3x⁵e^4x/8)+(15x⁴e^4x/32)-(15x³e^4x/32)+(45x²e^4x/128)-(45xe^4x/256)+(45e^4x/1024)+c

400

∫(x⁴cos(9x))dx

(x⁴sin(9x)/9)+(4x³cos(9x)/81)-(4x²sin(9x)/243)-(8xcos(9x)/2187)+(8sin(9x)/19683)+c

500

∫(e^2xsin(4x))dx

(-⅕)e^2xcos(4x)+(1/10)e^2xsin(4x)+c

Volume of Cylindrical Shells

100

Find volume of a solid generated by revolving f=x^2 bounded by x=3, x=0, and y=0 around the y-axis

81pi/2

200

Find the volume of the solid made by revolving the region bounded by x=(y-4)², the x-axis, and the y-axis, around the x-axis

128pi/3

300

Find the volume of the solid created when function y=-x⁴+16 in the 1st quadrant is revolved around x=-3.

2944pi/15

400

Find the volume of a solid generated by rotating the region in quadrant 1 bounded by x=12y-3y² and x=0 about the x-axis

128pi

500

Find the volume of the solid generated by rotating the region between x=2y2 and x=6y around the x-axis

27 pi