Integral Test Review

Definite Integrals

100

d/dxint_0^xsqrttdt

sqrtx

200

d/dxint_(5x)^6costdt

-5cos(5x)

300

d/dxint_(2x)^(x^2)lnwdw

2xlnx^2-2ln2x

400

int_2^4(x+2)dx

10

500

int_2^3xe^(x^2)dx

(e^9-e^4)/2

Fractions

100

int(x^3+7x-1)/x dx

x^3/3+7x-lnabsx+C

200

int(2x)/(x^2+7)^4dx

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

300

int(x+3)/(x^2+6x)dx

1/2lnabs(x^2+6x)+C

400

int(sec^2x)/tanxdx

lnabstanx+C

500

inttanxdx

-lnabscosx+C

Equations

100

int(x^3+3x^2-6)dx

x^4/4+x^3-6x+C

200

int4e^(2x)dx

2e^(2x)+C

300

int(1/x+1/(x^2)+sqrtx)dx

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

400

intxsec^2(6x^2)dx

1/12tan(6x^2)+C

500

intx^2cos(x^3)sin^6(x^3)dx

1/21sin^7(x^3)+C

Harder Equations

100

Given y'=2x+6. Find y= if y(2)=3 

x^2+6x-13

200

Find the average of the following on the interval [0, 2]

y=x^2+1

14/6 =7/3

300

intx^2sinxdx

-x^2cosx+2xsinx+2cosx+C

400

inte^(3x)sinxdx

(-e^(3x)cosx+3e^(3x)sinx)/10+C

500

int(9x)/(x^2+x-2)dx

3lnabs(x-1)+6lnabs(x+2)+C