Math / Calculus

QuestionFind the general antiderivative of the function $f(x) = 2x^3 - \frac{2}{3}x^2 + 5x$ .

Studdy Solution

STEP 1

1. We are given the function $f(x) = 2x^3 - \frac{2}{3}x^2 + 5x$ .
2. We need to find the general antiderivative of this function.
3. The general antiderivative will include a constant of integration, $C$ .

STEP 2

1. Identify the antiderivative of each term in the function.
2. Integrate each term separately.
3. Combine the integrated terms and include the constant of integration.

STEP 3

Identify the antiderivative of each term in the function:
- The antiderivative of $2x^3$ is $\frac{2}{4}x^4$ . - The antiderivative of $-\frac{2}{3}x^2$ is $-\frac{2}{3} \cdot \frac{1}{3}x^3$ . - The antiderivative of $5x$ is $\frac{5}{2}x^2$ .

STEP 4

Integrate each term separately:
- For $2x^3$ , the antiderivative is $\frac{2}{4}x^4 = \frac{1}{2}x^4$ . - For $-\frac{2}{3}x^2$ , the antiderivative is $-\frac{2}{9}x^3$ . - For $5x$ , the antiderivative is $\frac{5}{2}x^2$ .

STEP 5

Combine the integrated terms and include the constant of integration:
The general antiderivative is:
$F(x) = \frac{1}{2}x^4 - \frac{2}{9}x^3 + \frac{5}{2}x^2 + C$
where $C$ is the constant of integration.
The general antiderivative of the function is:
$F(x) = \frac{1}{2}x^4 - \frac{2}{9}x^3 + \frac{5}{2}x^2 + C$

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