FTC
the derivative of ∫ f(t)dt from a to x
f(x)
∫(1-2x)^9dx
-1/20(1-2x)^10+C
area between y = x ^3 and y = x^ 2
1/12
volume of y = x+1 y=0 from x=0 to x=2 about the x-axis
26pi/3
∫(xe^2x)dx
((1/2)xe^2x)-((1/4)e^2x)+C
find the derivative of ∫ (t + t^3)dt from 0 to x
x + x^3
∫cos(2x)dx
area between y=e^x and y=(x^2)-1 from x=-1 to 1
e-(1/e)+4/3
volume b/w y=x^2 and y=2-x^2 rotated about x-axis
16pi/3
∫xcos(5x)dx
(1/5)xsin(5x)+(1/25)cos(5x)+C
find the derivative of ∫ (t-t^2)^8 from 5 to s
(s-s^2)^8
∫sin^2(x)cos^3(x)dx
(1/3)sin^3(x)-(1/5)sin^5(x)+C
area between y=(x-2)^2 and y=x
9/2
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
∫(x^2(e^x))dx
(x^2)(e^x)-2x(e^x)+2(e^x)+C
find the derivative of ∫ (e^(t^2))dt from x^2 to x^3
3(x^2)(e^(x^6))-2x(e^(x^4))
∫tan^3(x)dx
(1/2)tan^2(x)-ln|sec(x)|+C
area between x=2y^2 and x=4+y^2
32/3
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
∫(x(lnx)^2)dx
((1/2)(x^2)(lnx)^2)-(1/2)(x^2)lnx+(1/4)(x^2)+C