f(x) = 6x^2 -8x +3
how would I find the most general antiderivative of the function. I also have to check my answer by differentiation.
Would I have to find the derivative first? (I got 12x -8)
What are the steps to solving a problem like this?
No, don't do the derivative first. Do that on the integral, to check your answer.
I use the word "integral" for what you are calling "antiderivative". You can say I am old-fashioned, but I don't see a need for a longer word.
The general rule for integrating any
a x^n term
is that the integral is
a x^(n+1)/(n+1)
When you have a polynomial, as you do here, the integral is the sum of the integrals of the terms.
The first term of the integral is
6 x^3/3 = 2 x^3. Take the derivative of that and you get the original 6 x^2 back, so it is the right answer.
Now you do the other terms, and add them up.
Remember that the most general integral has an arbitrary constant term added.
To find the most general antiderivative of the function f(x) = 6x^2 - 8x + 3, you can follow these steps:
1. Apply the power rule for integration to each term individually. The power rule states that the integral of a x^n term is (a/(n+1)) * x^(n+1).
2. For the first term, 6x^2, you can apply the power rule to get (6/(2+1)) * x^(2+1) = 2x^3.
3. For the second term, -8x, you can apply the power rule to get (-8/(1+1)) * x^(1+1) = -4x^2.
4. For the third term, 3, you can consider it as 3x^0 and apply the power rule to get (3/(0+1)) * x^(0+1) = 3x.
5. Now, you can sum up the results from steps 2-4 to obtain the most general antiderivative:
2x^3 - 4x^2 + 3x + C, where C is the constant of integration.
To check your answer by differentiation:
1. Take the derivative of 2x^3 - 4x^2 + 3x + C.
2. The derivative of 2x^3 is 6x^2, which matches the original first term of the function (6x^2).
3. The derivative of -4x^2 is -8x, which matches the original second term of the function (-8x).
4. The derivative of 3x is 3, which matches the original third term of the function (3).
5. As for the constant term, the derivative of any constant is always 0, so the C term disappears when taking the derivative.
By following these steps, you can find the most general antiderivative of the function and check your answer by differentiation.