Calculus Examples

$f (x) = 5 x - 6$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int 5 x - 6 d x$

Step 3

Split the single integral into multiple integrals.

$\int 5 x d x + \int - 6 d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int x d x + \int - 6 d x$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$5 (\frac{1}{2} x^{2} + C) + \int - 6 d x$

Step 6

Apply the constant rule.

$5 (\frac{1}{2} x^{2} + C) - 6 x + C$

Step 7

Simplify.

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Step 7.1

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

$5 (\frac{x^{2}}{2} + C) - 6 x + C$

Step 7.2

Simplify.

$\frac{5 x^{2}}{2} - 6 x + C$

Step 7.3

Reorder terms.

$\frac{5}{2} x^{2} - 6 x + C$

Step 8

The answer is the antiderivative of the function $f (x) = 5 x - 6$ .

$F (x) =$ $\frac{5}{2} x^{2} - 6 x + C$