Calculus Examples

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

Step 1

Write $2 x + \frac{3}{x^{2}}$ as a function.

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the expression.

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Step 8.1

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

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

Step 8.2

Simplify.

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Step 8.2.1

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

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

Step 8.2.2

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

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Step 8.2.2.1

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

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

Step 8.2.2.2

Multiply $2$ by $- 1$ .

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Multiply $- 1$ by $3$ .

$x^{2} - 3 x^{- 1} + C$

Step 11

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

$F (x) =$ $x^{2} - 3 x^{- 1} + C$