Find an Antiderivative

Question

Find an antiderivative F(x) with F′(x)=f(x)=4+15x2+15x4 and F(1)=0.

 

Remember to include a "+ C" if appropriate.

 

F(x)=

 

 __________

 

Answer 1

Question: Find an antiderivative F(x) with F′(x)=f(x)=4+15x²+15x⁴ and F(1)=0. Remember to include a "+ C" if appropriate.

Answer: We don't have specified bounds of integration, so this is an indefinite integral. We will need to include C before we find the function F(x) such that F(1) = 0. The value of C that makes this a reality will be in the final answer, and no letter C will be in the final answer. C is meant to suggest a family of antiderivatives. We just need to single out one member of the family.

Integration is a linear operator, so let's work our way across:

4: The antiderivative of a constant is that constant times the independent variable (here, x). So we get 4x.

15x^2: Bring the constant out in front such that we'll do 15 * ∫x² dx. By power rule, we'll raise the exponent by 1 and divide the newly modified integrand by the new exponent. Since 2+1 = 3, we'll have x³/3. Don't forget to multiply that by 15 to get 5x³.

15x⁴: Likewise, bring the constant out in front such that we'll do 15 * ∫x⁴ dx. By power rule, we'll raise the exponent by 1 and divide the newly modified integrand by the new exponent. Since 4+1 = 5, we'll have x⁵/5. Don't forget to multiply that by 15 to get 3x⁵.

The indefinite integral of f(x)=4+15x²+15x⁴ is:

F(x) = 4x + 5x³ + 3x⁵ + C.

But we still need to figure out the C that will make certain that when we plug in 1 wherever we see x, F(x) will evaluate to 0. Let's figure that out now:

(0) = 4(1) + 5(1)³ + 3(1)⁵ + C -----> 0 = 4 + 5 + 3 + C -----> C = -12

The final model is:

F(x) = 4x + 5x³ + 3x⁵ - 12