find an antiderivative

Question

Find an antiderivative F(x) with F′(x)=f(x)=8+15x2+21x6F'x=fx=8+15x2+21x6 and F(1)=0F1=0.

 

Remember to include a "+ C" if appropriate.

 

F(x)=

 

Answer 1

I am assuming you mean f(x) = 8 + 15x² + 21x⁶ with F'(x) = f(x) and F(1) = 0

The integral (anti-derivative) of power functions (axⁿ) is axⁿ⁺¹/(n+1) (n not equal to -1 and for constant term, treat as ax⁰) (You can see this by taking the derivative of the integral and getting back the function. Anti-derivatives are linear, so constants multiply and addition and subtraction will still hold after the integral:

F(x) = 8x + (15/3)x³ + (21/7)x⁷ + C = 8x +5x³ +(21/7)x⁷ + C

Now use the condition F(1)=0

8 + 5 + 21/7 +C = 0 (solve for C)

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