Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.)
f(x) = 1/2x+ 2/3x^2 - 5/6x^3
what is .25x^2 + (2/9)x^3- (5/24)x^4 + C
check that.
x / 2 + 2 x^2 / 3 - 5 x^3 / 6 =
3 ∙ x / 3 ∙ 2 + 2 ∙ 2 x^2 / 2 ∙ 3 - 5 x^3 / 6 =
3 ∙ x / 6 + 4 x^2 / 6 - 5 x^3 / 6 =
( 1 / 6 ) ∙ ( 3 x + 4 x^2 - 5 x^3 ) =
( 1 / 6 ) ∙ ( - 5 x^3 + 3 x + 4 x^2 )
∫ ( 1 / 6 ) ∙ ( - 5 x^3 + 4 x^2 + 3 x ) dx =
( 1 / 6 ) ∙ ( - 5 ∫ x^3 dx + 4 ∫ x^2 dx + 3 ∫ x dx ) =
( 1 / 6 ) ∙ ( - 5 x^4 / 4 + 4 x^3 / 3 + 3 x^2 ) + C =
( 1 / 6 ) ∙ ( - 5 ∙ 3 x^4 / 3 ∙ 4 + 4 ∙ 4 x^3 / 4 ∙ 3 + 3 ∙ 12 x^2 / 12 ) + C =
( 1 / 6 ) ∙ ( - 15 ∙ x^4 / 12 + 16 x^3 / 12 + 36 x^2 / 12 ) + C =
( 1 / 6 ∙ 12 ) ∙ ( - 15 ∙ x^4 + 16 x^3 + 36 x^2 ) + C =
( 1 / 72 ) ∙ ( - 15 ∙ x^4 + 16 x^3 + 36 x^2 ) + C =
( - 1 / 72 ) ∙ ( 15 ∙ x^4 - 16 x^3 - 36 x^2 ) + C
The same answer as bobpursley's but written differently.
To find the most general antiderivative of the function f(x) = 1/2x + 2/3x^2 - 5/6x^3, we will apply the power rule of integration. The power rule states that for any term of the form x^n, where n is any real number except -1, the antiderivative is given by (1/(n+1))x^(n+1) + C, where C is the constant of the antiderivative.
Let's determine the antiderivative of each term individually and then sum them up:
1/2x: The power rule tells us that the antiderivative will be (1/(1+1))x^(1+1) + C = (1/2)x^2 + C.
2/3x^2: The power rule tells us that the antiderivative will be (1/(2+1))x^(2+1) + C = (1/3)x^3 + C.
-5/6x^3: The power rule tells us that the antiderivative will be (1/(3+1))x^(3+1) + C = (-5/24)x^4 + C.
Now, let's sum up all the antiderivatives we found:
(1/2)x^2 + (1/3)x^3 - (5/24)x^4 + C.
This is the most general antiderivative of the function f(x).
To verify our answer, we can differentiate it and check if it matches the original function. The derivative of (1/2)x^2 + (1/3)x^3 - (5/24)x^4 + C with respect to x will indeed give us 1/2x + 2/3x^2 - 5/6x^3, which is the original function f(x).