Integral Review

Properties of Integrals

100

int [f(x)+g(x)]dx

intf(x)dx + intg(x)dx

200

intcf(x)dx

cintf(x)dx

300

int_a^af(x)dx

0

400

int_a^bf(x)dx + int_b^cf(x)dx

int_a^cf(x)dx

500

int_-pi^pi(tan(x^3)+x^2sinx)dx

[Hint: is the function even or odd?]

0

Fundamental Theorem of Calculus, Part 1

100

d/dxint_0^x(t+1)dt

x+1

200

d/dxint_pi^x(tsint)/(t-1)dt

(xsinx)/(x-1)

300

d/dxint_x^-1ln(t^2-1)dt

-ln(x^2-1)

400

d/dxint_0^(2x^2+x)t(t-2)^2dt

(2x^2+x)(2x^2+x-2)^2(4x+1)

500

d/dxint_sinx^-2te^(t^2)dt

-sinxe^(sin^2x)cosx

Fundamental Theorem of Calculus, Part 2

100

int_0^2 3x^2dx

8

200

int_-1^1 e^xdx

e^1-e^-1

300

int_(-pi/2)^(pi/2)cosxdx

2

400

int_0^1 1/(1+x^2)dx

pi/4

500

int_1^2(1/x^2+1/x+1)dx

3/2+ln2

Basic Integrals

100

int(x^3-2x^2+x+3)dx

x^4/4-(2x^3)/3+x^2/2+(3x)/1+C

200

int(sqrt(x)+x^(-2)+root(3)x^4)dx

2/3x^(3/2)+x^-1/-1+3/7x^(7/3)+C

300

int(2/x+4e^x)dx

2lnabsx+4e^x+C

400

int cscxcotxdx

-cscx+C

500

int 4/sqrt(1-x^2)dx

4sin^-1x+C

Integrals w/ Substitution

100

int(2x+3)^5dx

(2x+3)^6/6*1/2+C

200

int(4x^2-2x+1)^-3(4x-1)dx

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

300

int2xsinx^2dx

-cosx^2+C

400

int 1/sqrt(16-x^2)dx

sin^-1(x/4)+C

500

int sec^2xtan^5xdx

tan^6x/6+C