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_(0)^(pi)sinxcos^2xdx

2/3

400

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

sqrt(2)-1

500

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

3/2+ln2

Basic Integrals

100

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

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

200

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

2/5x^(5/2)-4/3x^(3/2)+2sqrt(x)+C

300

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

2lnabsx+4e^x+C

400

int cscxcotxdx

-cscx+C

500

int (4x^2+3x-10)/(x+3)dx

2X^2-9x+17lnabs(x+3)

Integrals w/ Substitution

100

int(x+3)/(x-5)^2dx

lnabs(x-5)+8/(x-5)+C

200

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

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

300

int2xsinx^2dx

-cosx^2+C

400

int 3xe^(2x^2-1)dx

3/4e^(2x^2-1)

500

int sec^2xtan^5xdx

tan^6x/6+C