Test 1 Review!!

FTC

100

the derivative of ∫ f(t)dt from a to x

f(x)

200

find the derivative of ∫ (t + t^3)dt from 0 to x 

x + x^3

300

find the derivative of ∫ (t-t^2)^8 from 5 to s

(s-s^2)^8

400

find the derivative of ∫ (e^(t^2))dt from x^2 to x^3

3(x^2)(e^(x^6))-2x(e^(x^4))

U-Sub + Trig Integrals

100

∫(1-2x)^9dx

-1/20(1-2x)^10+C

200

∫cos(2x)dx

(1/2)sin(2x)+C

300

∫sin^2(x)cos^3(x)dx

(1/3)sin^3(x)-(1/5)sin^5(x)+C

400

∫tan^3(x)dx

(1/2)tan^2(x)-ln|sec(x)|+C

Area b/w Curves

100

area between y = x ^3 and y = x^ 2

1/12

200

area between y=e^x and y=(x^2)-1 from x=-1 to 1

e-(1/e)+4/3

300

area between y=(x-2)^2 and y=x

9/2

400

area between x=2y^2 and x=4+y^2

32/3

Volume

100

volume of y = x+1 y=0 from x=0 to x=2 about the x-axis 

26pi/3

200

volume b/w y=x^2 and y=2-x^2 rotated about x-axis

16pi/3

300

The triangular region bounded by the lines y = x, the x-axis and x = 1 is rotated about the vertical line x = −1. Find the volume.

5pi/3

400

The area between y = x ^2 and the x-axis from x = 0 to x = 1 is the base of a solid whose cross-sections perpendicular to the x-axis are squares. Find the volume

1/5

Integration by Parts

100

∫(xe^2x)dx

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

200

∫xcos(5x)dx

(1/5)xsin(5x)+(1/25)cos(5x)+C

300

∫(x^2(e^x))dx

(x^2)(e^x)-2x(e^x)+2(e^x)+C

400

∫(x(lnx)^2)dx

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