Integration Techniques

IBP

100

∫ xe^x dx

e^x(x-1)+C

200

∫ ln(x) dx

xln(x)-x+C

300

∫ x² cos(x) dx

x²sin(x)+2xcos(x)-2sin(x)+C

400

Using the bounds from 0 to 1, evaluate ∫ xln(x) dx

-1/4

500

Using the bounds from 0 to pi, evaluate ∫ xsin(x) dx

pi

U-Sub

100

∫(2x)·cos(x²) dx

sin(x²)+C

200

∫1/(1+(3x+2)²) dx

(1/3)arctan(3x+2)+C

300

∫ x·√(x²+1) dx

(1/3)(x²+1)^3/2+C

400

Using the bounds from 1 to 2, evaluate ∫ x/(x²+1) dx

(1/2)ln(5/2)

500

Using the bounds from 0 to 1, evaluate ∫ x³/(x⁴+1)² dx

1/8

Random

100

∫1/x dx

ln|x|+C

200

∫ 5x·cos(x²) dx

(5/2)sin(x²) +C

300

∫ 1/√(9-x²) dx

arcsin(x/3)+C

400

Using the bounds from 2 to 3, evaluate ∫ 1/(x²-1) dx

-(1/2)ln(2)

500

Using the bounds from 0 to 1, evaluate ∫xe^{x^2} dx

(1/2)(e-1)

Partial Fractions

100

∫3/(x²-4) dx

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

200

∫(x+3) / ((x+1)(x-2)) dx

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

300

∫(2x²+3x+1) / ((x+1)(x²-1))dx

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

400

Using the bounds from 2 to 3, evaluate ∫(5x+1)/(x² - 1) dx

7ln(2)-2ln(3)

500

Using the bounds from 0 to 1, evaluate ∫(x³+2x²+3x+4) / ((x²+1)(x+1))dx

1+ln(2)+(pi/2)

Trig Sub

100

∫1/(x²√(x²-4)) dx

(1/4)((√x²-4)/x)+ C

200

∫1/√(16-x²)dx

arcsin(x/4)+C

300

∫1 / (x²√(x²-16)) dx

-(√(x²-16)) / (16x²)+C

400

Using the bounds from 0 to 2, evaluate ∫x² / (√4-x²) dx

4pi/3

500

Using the bounds from 2 to 3, evaluate ∫x³ / (x² - 4)^3/2 dx

((3√5)/2) + (5/2)ln((3+√5)/2)