Calculus Examples

$R' (x) = 600 - 0.6 x$

Step 1

The function

$R (x)$ can be found by evaluating the indefinite integral of the derivative

$R' (x)$ .

$R (x) = \int R' (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 600 d x + \int - 0.6 x d x$

Step 3

Apply the constant rule.

$600 x + C + \int - 0.6 x d x$

Step 4

Since

$- 0.6$ is constant with respect to

$x$ , move

$- 0.6$ out of the integral.

$600 x + C - 0.6 \int x d x$

Step 5

By the Power Rule, the integral of

$x$ with respect to

$x$ is

$\frac{1}{2} x^{2}$ .

$600 x + C - 0.6 (\frac{1}{2} x^{2} + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

$600 x - 0.6 (\frac{1}{2}) x^{2} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine

$- 0.6$ and

$\frac{1}{2}$ .

$600 x + \frac{- 0.6}{2} x^{2} + C$

Step 6.2.2

Move the negative in front of the fraction.

$600 x - \frac{0.6}{2} x^{2} + C$

Step 6.3

Simplify.

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Step 6.3.1

Divide

$0.6$ by

$2$ .

$600 x - 1 \cdot 0.3 x^{2} + C$

Step 6.3.2

Multiply

$- 1$ by

$0.3$ .

$600 x - 0.3 x^{2} + C$

Step 7

The function

$R$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$R (x) = 600 x - 0.3 x^{2} + C$