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f(x) = x^2 + 5 f’(x) = 2x
f(x) is an antiderivative of 2x
g(x) = x^2 - 3 g’(x) = 2x
how do i know what the antiderivative is?
actual antiderivative of g’(x) or f’(x) is: x^2 + c
f(x) =x^3 + x^2 + 50 g(x) = x^3 + x^2 - 80
both are antiderivatives of 3x^2 + 2x
integral sign
( find the antiderivative )
antiderivative
differential ( what variable to use )
3x
(1/2)x^6 + c
t^6 - t^4 + 10/3t^3 + t + c
5ln(|x|) + c
(x^(n+1))/(n+1) + C
(x^(n+1))/(n+1) + C
3x^6 / 6 +c 3/6x^6 +c 1/2x^6 +c im goated
alternatively, if i try to use power rule
5x^-1 5x^0/0 = no good
y^-1/2 (y^(-1/2+1))/(-1/2 + 1) + c (y^1/2)/(1/2) + c
2y^(1/2) + c 2sqrt(y) + c
e^x - 5/7x^7/5 + 3x + c
t + 1/t + 1/t^2 t + 1/t + t^-2 (1/2)t^2 + ln|t| + t^-1 / -1 + c
(1/2)t^2 + ln|t| - t^-1 + c
x - 1/x
1/2x^2 - ln(|x|) + c
1/2(1)^2 - ln(|1|) + c = 3/2 (1/2) - ln(1) + c = 3/2 .5 - ln(1) + c = 3/2 .5 + c = 1.5 c = 1
4e^x + 5x
4e^x + 5/3x^3 + c
-4 = 4e^(0) + 5/3(0)^3 + c -4 = 4 + c -8 = c
so
4e^x + 5/3x^3 - 8
-5e^t - 8t -5e^t - 4t^2
-9/8 -9/8 x + c
5x^8 - 4 5/9x^9 - 4x + c
find antiderivative of 12/x^2 so thats 12x
so 12/-1x^-1 -12x
so it has to be a -12/x one
differentiate -7e^(x^2) -7e^(x^2) * 2x -14xe^(x^2)
so it has to be a -7e^(x^2) one so there is only one option.
150t - 6t^2 find antiderivative ( velocity is derivative of distance ) (150/2)t^2 - 6/3 t^3 75t^2 - 2t^3 + c
75(0)^2 - 2(0)^3 + c = 0 0 - 0 + c = 0 c = 0 so its just 75t^2 - 2t^3
75(10)^2 - 2(10)^3 5500
x^6(7x+5) 7x^7 + 5x
now integrate
7/8x^8 + 5/7x^7 + c
(7x-3)^2 (7x-3)(7x-3) 49x^2 -21x -21x + 9 49x^2 -42x + 9 now integrate
(49/3)x^3 + (-42/2)x^2 + 9x 49/3 x^3 - 21x^2 + 9x
-7x^6 - 3/x - 4/x
-7x^6 - 3x^-1 - 4x
-7/7x^7 - 3ln(x) - (4/4)/x^-4 -x^7 - 3ln(x) - x^-4 -x^7 - 3ln(x) - 1/x^4
-7sqrt(x) + 5qdrt(x) -7x^(1/2) + 5x^(1/4) -7/(3/2) x^(3/2) + (5)/(5/4) x^(5/4) (-7)(2/3)x^(3/2) + (5)(4/5)x^(5/4) (-14/3)x^(3/2) + (20/5)x^(5/4) (-14/3)x^(3/2) + 4x^(5/4) + c
40-10t^(1/2)
40t- (10/(3/2))t^(3/2) 40t - (10)(2/3)t^(3/2) 40t - (20/3)t^(3/2) + c 40(0) - (20/3)(0)^(3/2) + c = 9300 0 - 0 + c = 9300 c = 9300
40t - 15t^(3/2) + 9300
150-.08x 150x - .08/2 x^2 150x - .04x^2
6x^(-1/3) - 7 (6)/(2/3)x^(2/3) - 7x (6)(3/2)x^(2/3) - 7x (18/2)x^(2/3) - 7x 9x^(2/3) - 7x
-5/x + 2x^(3/5) - 3e^x -5x^-1 + 2x^(3/5) - 3e
integrate -5ln|x| + (2/(8/5))x^(8/5) - 3e^x -5ln|x| + (2)(5/8)x^(8/5) - 3e^x -5ln|x| + 10/8x^8/5 -3e^x + c -5ln|x| + (10/8)x^(8/5) -3e^x + c -5ln|x| + (5/4)x^(8/5) -3e^x + c
4x^2/x^2 - 7x/x^2 - 3/x^2 4 - 7x/x^2 - 3/x^2 4 - 7x(x^-2) - 3(x^-2) 4 - 7x^-1 - 3x^-2 integrate!
4x - 7ln(x) + 3x^-1 + c 4x - 7ln(x) + 3/x + c
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