Calculus Examples

$f (x) = \frac{2}{x} + \frac{3}{x^{2}}$

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.

$F (x) = \int \frac{2}{x} + \frac{3}{x^{2}} d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \frac{1}{x} d x + \int \frac{3}{x^{2}} d x$

Step 5

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

$2 (\ln (| x |) + C) + \int \frac{3}{x^{2}} d x$

Step 6

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$2 (\ln (| x |) + C) + 3 \int \frac{1}{x^{2}} d x$

Step 7

Apply basic rules of exponents.

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Step 7.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$2 (\ln (| x |) + C) + 3 \int {(x^{2})}^{- 1} d x$

Step 7.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 7.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 (\ln (| x |) + C) + 3 \int x^{2 \cdot - 1} d x$

Step 7.2.2

Multiply $2$ by $- 1$ .

$2 (\ln (| x |) + C) + 3 \int x^{- 2} d x$

Step 8

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

$2 (\ln (| x |) + C) + 3 (- x^{- 1} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$2 \ln (| x |) + 3 (- \frac{1}{x}) + C$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $- 1$ by $3$ .

$2 \ln (| x |) - 3 \frac{1}{x} + C$

Step 9.2.2

Combine $- 3$ and $\frac{1}{x}$ .

$2 \ln (| x |) + \frac{- 3}{x} + C$

Step 9.2.3

Move the negative in front of the fraction.

$2 \ln (| x |) - \frac{3}{x} + C$

Step 10

The answer is the antiderivative of the function $f (x) = \frac{2}{x} + \frac{3}{x^{2}}$ .

$F (x) =$ $2 \ln (| x |) - \frac{3}{x} + C$