Cal II: Integrals

Substitution

100

2xe(^{x^2})

e^{(x^2)}+C

200

(x³)/(1+x⁴)

(ln(x⁴+1))/4+C

300

(3sin(x))/sqrt(1+cos(x))

-6(sqrt(cos(x)+1))+C

400

e^xsin(3e^x-1)

-(cos(3e^x-1)/3)+C

500

(3+sec(x)tan(x))(6x+2sec(x))⁴

(2sec(x)+6x)⁵/10+C

By Parts

100

xe^x

(x-1)e^x+C

200

3xsin(5x)

(3(sin(5x)-8xcos(5x))/25+C

300

x²cos(2x)

((4x²-2)sin(2x)+4xcos(2x))/8+C

400

x⁴ln(x)

((x⁵ln(x))/5)-(x⁵/25)+C

500

ln(x)

x(ln(x)-1)+C

Trig

100

sin³(x)cos²(x)

(cos⁵(x)/5)-(cos³(x)/3)+C

200

sin⁵(5x)cos³(5x)

(sin⁶(5x)/30)-(sin⁸(5x)/40)+C

300

sin²(3x)

-(sin(6x)-6x)/12+C

400

sin²(x)cos²(x)

-(sin(4x)-4x)/32+C

500

cos³(17x)sin⁴(17x)

(sin⁵(17x)/85)-(sin⁷(17x)/119)+C

Trig Sub

100

1/(x²sqrt(x²+9))

-(sqrt(x²+9))/(9x)+C

200

sqrt(4-x²)

x(sqrt(1-x²/4)+2arcsin(x/2)+C

300

1/(sqrt(25x²-4))

(ln(sqrt(25x²-4)+5x))/5+C

400

(x⁵)/(36x²+1)^(3/2)

(162x⁴-18x²-1)/(17496(sqrt(36x²+1)))+C

500

x/sqrt(5-4x-x²)

-2arcsin((x+2)/3)-sqrt(-(x-1)(x+5))+C

Partial Fractions

100

(x-1)/(x²-4x+4)

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

200

(2x+2)/(x²+x-2)

ln(x²+x-2)+(ln(x-1)-ln(x+2))/3+C

300

(3x+2)/(x²-2x+5)

(3ln(x²-2x+5))/2)+(5arctan((x-1)/2))/2+C

400

(x³+1)/(x²-5x+6)

((x²+10x)/2)-9ln(x-2)+28ln(x-3)+C

500

(x⁴-5x³+6x²-18)/(x³-3x²)

((4lnx+(x-4)x-4ln(x-3))/2)-(6/x)+C