Question

Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.)
f ''(x) = 8 + x3 + x5

Answer 1

Task

Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.)

$$f''(x) = 8 + x^3 + x^5$$

Solution

Find $f'(x)$:

$$f'(x) = \int \left(8 + x^3 + x^5\right) dx = \int 8dx + \int x^3 dx + \int x^5 dx = 8x + \frac{x^4}{4} + \frac{x^6}{6} + C$$

Find $f(x)$:

$$\begin{aligned} f(x) &= \int f'(x) dx = \int \left(8x + \frac{x^4}{4} + \frac{x^6}{6} + C\right) dx = \int 8x dx + \int \frac{x^4}{4} dx + \int \frac{x^6}{6} dx + \int C dx \\ &= 8\frac{x^2}{2} + \frac{x^5}{4 \times 5} + \frac{x^7}{6 \times 7} + Cx + D = 4x^2 + \frac{x^5}{4 \times 5} + \frac{x^7}{6 \times 7} + Cx + D \end{aligned}$$

Answer

$$f(x) = 4x^2 + \frac{x^5}{20} + \frac{x^7}{42} + Cx + D$$