Indefinite Integrals
2x
x^2 + C
dy/dx = 2, y(0) = -3
y = 2x -3
What is the difference between the right Riemann sum and the left Riemann sum?
Using different endpoints to calculate the height of a sub-interval's rectangle (the left vs. right end-point)
What is the derivative of the integral from 0 to x of (t^2 + 3)dt
x^2 + 3
What is the equation this part of FTC?
the integral from a to b of f(x)dx = F(b) - F(a)
4x^6 - 2x^3 + 7x - 4
(4/7)x^7 - (1/2)x^4 + (7/2)x^2 - 4x + C
dy/dx = 6x - 3, y(0) = 4
y = 3x^2 - 3x + 4
Left Riemann sum using n = 4 subintervals for y = x^2 + 1 on [0,2]
3.75
The derivative of the integral from 2 to x of 13/(1 + t^2) dt
13/(1 + x^2)
the integral from 1 to 4 of (2x + 3)dx
24
(4/x^2) + 2 - 1/(8x^3)
-4x^-1 + 2x + (1/16)x^-2 + C
dy/dx = sin(x), y(0) = 5
y = -cos(x) + 6
Left Riemann sum using n = 5 subintervals for f(x) = x^3 = 5x^2 + 6x + 5.
22.56
The derivative of the integral of 4 to x of [9cos^2(t^2 - 6t + 1)]dt
9cos^2(x^2 - 6x + 1)
The integral of 2 to 1 of (2x^3 - 6x^2)/(x^2) dx
6/x^3 - 2/x
-3x^-2 - 2ln|x| + X
dy/dx = 3/(x^2), y(1) = 2
y = -3/x + 5
Right Riemann sum using n = 4 subintervals for y = x^2 + 1 on [0,2]
5.75
The derivative of the integral of 1 to x^1/2 of sin(t)dt
sin(x^1/2)/(2x^1/2)
The integral from 1 to 4 of 8/(x^1/2) - 12x^3/2 dt
-664/5
(x^4 - x^1/3)/(6x^1/2)
(1/27)x^(9/2) - (1/5)x^(5/6) + C
dy/dx = 2xy, y(0) = 3
y = 3e^(x^2)
2 + pi
The derivative of the integral of x to 2x of t^3dt
15x^3
The integral of 1 to 9 of (x - 1)/x^(1/2)
40/3