Calculus Examples

$f (x) = \frac{2 x^{4} + 3}{x^{2}}$

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.

$F (x) = \int \frac{2 x^{4} + 3}{x^{2}} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

$\int (2 x^{4} + 3) {(x^{2})}^{- 1} d x$

Step 3.2

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

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Step 3.2.1

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

$\int (2 x^{4} + 3) x^{2 \cdot - 1} d x$

Step 3.2.2

Multiply $2$ by $- 1$ .

$\int (2 x^{4} + 3) x^{- 2} d x$

Step 4

Multiply $(2 x^{4} + 3) x^{- 2}$ .

$\int 2 x^{4} x^{- 2} + 3 x^{- 2} d x$

Step 5

Multiply $x^{4}$ by $x^{- 2}$ by adding the exponents.

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Step 5.1

Move $x^{- 2}$ .

$\int 2 (x^{- 2} x^{4}) + 3 x^{- 2} d x$

Step 5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 2 x^{- 2 + 4} + 3 x^{- 2} d x$

Step 5.3

Add $- 2$ and $4$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Simplify.

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Step 11.2.1

Combine $2$ and $\frac{1}{3}$ .

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

Step 11.2.2

Multiply $- 1$ by $3$ .

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

Step 12

The answer is the antiderivative of the function $f (x) = \frac{2 x^{4} + 3}{x^{2}}$ .

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