Calculus Examples

$\frac{2}{5} - \frac{7}{x}$

Step 1

Write $\frac{2}{5} - \frac{7}{x}$ as a function.

$f (x) = \frac{2}{5} - \frac{7}{x}$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int \frac{2}{5} - \frac{7}{x} d x$

Step 4

Split the single integral into multiple integrals.

$\int \frac{2}{5} d x + \int - \frac{7}{x} d x$

Step 5

Apply the constant rule.

$\frac{2}{5} x + C + \int - \frac{7}{x} d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{2}{5} x + C - \int \frac{7}{x} d x$

Step 7

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$\frac{2}{5} x + C - (7 \int \frac{1}{x} d x)$

Step 8

Multiply $7$ by $- 1$ .

$\frac{2}{5} x + C - 7 \int \frac{1}{x} d x$

Step 9

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{2}{5} x + C - 7 (\ln (| x |) + C)$

Step 10

Simplify.

$\frac{2}{5} x - 7 \ln (| x |) + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{2}{5} - \frac{7}{x}$ .

$F (x) =$ $\frac{2}{5} x - 7 \ln (| x |) + C$