Calculus Examples

$(\frac{5}{8} x^{4} - 3 x^{2} + 2 x^{- 1}) d x$

Step 1

Write $(\frac{5}{8} x^{4} - 3 x^{2} + 2 x^{- 1}) d x$ as a function.

$f (x) = (\frac{5}{8} x^{4} - 3 x^{2} + 2 x^{- 1}) d x$

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.

$F (x) = \int (\frac{5}{8} x^{4} - 3 x^{2} + 2 x^{- 1}) d x d x$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine $\frac{5}{8}$ and $x^{4}$ .

$\int (\frac{5 x^{4}}{8} - 3 x^{2} + 2 x^{- 1}) d x d x$

Step 4.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int (\frac{5 x^{4}}{8} - 3 x^{2} + 2 \frac{1}{x}) d x d x$

Step 4.1.3

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

$\int (\frac{5 x^{4}}{8} - 3 x^{2} + \frac{2}{x}) d x d x$

Step 4.2

Apply the distributive property.

$\int (\frac{5 x^{4}}{8} d - 3 x^{2} d + \frac{2}{x} d) x d x$

Step 4.3

Simplify.

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Step 4.3.1

Combine $\frac{5 x^{4}}{8}$ and $d$ .

$\int (\frac{5 x^{4} d}{8} - 3 x^{2} d + \frac{2}{x} d) x d x$

Step 4.3.2

Combine $\frac{2}{x}$ and $d$ .

$\int (\frac{5 x^{4} d}{8} - 3 x^{2} d + \frac{2 d}{x}) x d x$

Step 4.4

Apply the distributive property.

$\int \frac{5 x^{4} d}{8} x - 3 x^{2} d x + \frac{2 d}{x} x d x$

Step 4.5

Simplify.

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Step 4.5.1

Multiply $\frac{5 x^{4} d}{8} x$ .

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Step 4.5.1.1

Combine $\frac{5 x^{4} d}{8}$ and $x$ .

$\int \frac{5 x^{4} d x}{8} - 3 x^{2} d x + \frac{2 d}{x} x d x$

Step 4.5.1.2

Raise $x$ to the power of $1$ .

$\int \frac{5 d (x^{4} x^{1})}{8} - 3 x^{2} d x + \frac{2 d}{x} x d x$

Step 4.5.1.3

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

$\int \frac{5 d x^{4 + 1}}{8} - 3 x^{2} d x + \frac{2 d}{x} x d x$

Step 4.5.1.4

Add $4$ and $1$ .

$\int \frac{5 d x^{5}}{8} - 3 x^{2} d x + \frac{2 d}{x} x d x$

Step 4.5.2

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

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Step 4.5.2.1

Move $x$ .

$\int \frac{5 d x^{5}}{8} - 3 (x \cdot x^{2}) d + \frac{2 d}{x} x d x$

Step 4.5.2.2

Multiply $x$ by $x^{2}$ .

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Step 4.5.2.2.1

Raise $x$ to the power of $1$ .

$\int \frac{5 d x^{5}}{8} - 3 (x^{1} x^{2}) d + \frac{2 d}{x} x d x$

Step 4.5.2.2.2

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

$\int \frac{5 d x^{5}}{8} - 3 x^{1 + 2} d + \frac{2 d}{x} x d x$

Step 4.5.2.3

Add $1$ and $2$ .

$\int \frac{5 d x^{5}}{8} - 3 x^{3} d + \frac{2 d}{x} x d x$

Step 4.5.3

Cancel the common factor of $x$ .

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Step 4.5.3.1

Cancel the common factor.

$\int \frac{5 d x^{5}}{8} - 3 x^{3} d + \frac{2 d}{x} x d x$

Step 4.5.3.2

Rewrite the expression.

$\int \frac{5 d x^{5}}{8} - 3 x^{3} d + 2 d d x$

Step 5

Split the single integral into multiple integrals.

$\int \frac{5 d x^{5}}{8} d x + \int - 3 x^{3} d d x + \int 2 d d x$

Step 6

Since $\frac{5 d}{8}$ is constant with respect to $x$ , move $\frac{5 d}{8}$ out of the integral.

$\frac{5 d}{8} \int x^{5} d x + \int - 3 x^{3} d d x + \int 2 d d x$

Step 7

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$\frac{5 d}{8} (\frac{1}{6} x^{6} + C) + \int - 3 x^{3} d d x + \int 2 d d x$

Step 8

Combine $\frac{1}{6}$ and $x^{6}$ .

$\frac{5 d}{8} (\frac{x^{6}}{6} + C) + \int - 3 x^{3} d d x + \int 2 d d x$

Step 9

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

$\frac{5 d}{8} (\frac{x^{6}}{6} + C) - 3 d \int x^{3} d x + \int 2 d d x$

Step 10

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{5 d}{8} (\frac{x^{6}}{6} + C) - 3 d (\frac{1}{4} x^{4} + C) + \int 2 d d x$

Step 11

Combine $\frac{1}{4}$ and $x^{4}$ .

$\frac{5 d}{8} (\frac{x^{6}}{6} + C) - 3 d (\frac{x^{4}}{4} + C) + \int 2 d d x$

Step 12

Apply the constant rule.

$\frac{5 d}{8} (\frac{x^{6}}{6} + C) - 3 d (\frac{x^{4}}{4} + C) + 2 d x + C$

Step 13

Simplify.

$\frac{5 d x^{6}}{48} - \frac{3 d x^{4}}{4} + 2 d x + C$

Step 14

Remove parentheses.

$\frac{5}{48} d x^{6} - \frac{3}{4} d x^{4} + 2 d x + C$

Step 15

The answer is the antiderivative of the function $f (x) = (\frac{5}{8} x^{4} - 3 x^{2} + 2 x^{- 1}) d x$ .

$F (x) =$ $\frac{5}{48} d x^{6} - \frac{3}{4} d x^{4} + 2 d x + C$