Calculus help question

Question

Find an antiderivative F(x) with F′(x)=f(x)=5+28x3+21x6 and F(1)=0.

 

Remember to include a "+ C" if appropriate.

Answer 1

Hi Prabh!

This is a great question. Specifically, it is testing your understanding of antiderivatives and indefinite integrals.

A little bit of background knowledge to help understand what you are doing in this question:

1. Antiderivative implies exactly what the name says. It is the "undoing" or the derivative, or going back to the original function, f(x), that you originally took its derivative. This is generally known as finding the indefinite integral.
2. F(x) = general antiderivative, indefinite integral
3. F'(x) = f(x) (the original function)
4. Integral of f(x) = F(x)
5. Additionally, when you find the general antiderivative (original function), you have to account for the possibility of there being a constant at the end of the original function that disappeared when you take the derivative. This is why we add a C on the end of each general antiderivative. For example:
6. 1) f(x) = x²+3, f'(x) = 2x
7. The derivative is the same as #2!
8. The antiderivative, F(x) = x²+C to account for the +3
9. 2) f(x) = x²+7, f'(x) = 2x
10. The derivative is the same as #1!
11. The antiderivative, F(x) = x²+C to account for the +7

Now, back to the question specifically.

1. One of the general antiderivative rules to find the original function is this (opposite of the power rule for derivatives):
2. The general antiderivative of xⁿ = (xⁿ⁺¹/(x+1)) + C.
3. The questions tells us that f(x) = 5 + 28x³ + 21x⁶. Therefore, using the rule provided above in #1, you will get the antiderivative of:
4. F(x) = 5x + 7x⁴ + 3x⁷ + C
5. This is the general antiderivative for this function.
6. Notice that F(x) still has a constant, C. The question gives us additional information in order to find the specific value of C for this antiderivative. It says that F(1) = 0 by setting F(x) equal to 0 and plugging in 1 for x. Therefore, to find C:
7. F(1) = 0 = 5(1) + 7(1)⁴ + 3(1)⁷ + C
8. 0 = 15 + C
9. C = -15
10. The final step for this question is to write out F(x) with all of the information that we have found in the previous steps. Therefore, the final answer should be:
11. F(x) = 5x + 7x⁴ + 3x⁷ - 15
12. This is a specific antiderivative for this function given the specific information about C in the question.

I hope this is helpful! If you need any more clarification, please let me know!

Answer 2

An antiderivative of xⁿ is xⁿ⁺¹ / (n+1) provided n ≠ -1. Secondly, an antiderivative for a sum of terms is the sum of the antiderivatives of each term. So 5x + 28(x⁴ / 4) + 21 (x⁷ / 7) = 5x + 7x⁴ + 3x⁷ is an antiderivative of f(x). Any two antiderivatives of f differ by a constant, so F(x) = 5x + 7x⁴ + 3x⁷ + C, where C is a constant. The condition F(1) = 0 implies 0 = 15 + C, so C = -15. Therefore, F(x) = 5x + 7x⁴ + 3x⁷ -15.