Calculus Examples

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

Step 1

Remove parentheses.

$\int \frac{4 x^{6} + 2 x^{4}}{x^{3}} d x$

Step 2

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.1

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

$\int (4 x^{6} + 2 x^{4}) x^{3 \cdot - 1} d x$

Step 3.2

Multiply $3$ by $- 1$ .

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

Step 4

Expand $(4 x^{6} + 2 x^{4}) x^{- 3}$ .

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Step 4.1

Apply the distributive property.

$\int 4 x^{6} x^{- 3} + 2 x^{4} x^{- 3} d x$

Step 4.2

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

$\int 4 x^{6 - 3} + 2 x^{4} x^{- 3} d x$

Step 4.3

Subtract $3$ from $6$ .

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

Step 4.4

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

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

Step 4.5

Subtract $3$ from $4$ .

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

Step 4.6

Simplify.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 7

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

$4 (\frac{1}{4} x^{4} + C) + \int 2 x d x$

Step 8

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

$4 (\frac{1}{4} x^{4} + C) + 2 \int x d x$

Step 9

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

$4 (\frac{1}{4} x^{4} + C) + 2 (\frac{1}{2} x^{2} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$4 (\frac{1}{4}) x^{4} + 2 (\frac{1}{2}) x^{2} + C$

Step 10.2

Simplify.

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Step 10.2.1

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

$\frac{4}{4} x^{4} + 2 (\frac{1}{2}) x^{2} + C$

Step 10.2.2

Cancel the common factor of $4$ .

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Step 10.2.2.1

Cancel the common factor.

$\frac{4}{4} x^{4} + 2 (\frac{1}{2}) x^{2} + C$

Step 10.2.2.2

Rewrite the expression.

$1 x^{4} + 2 (\frac{1}{2}) x^{2} + C$

Step 10.2.3

Multiply $x^{4}$ by $1$ .

$x^{4} + 2 (\frac{1}{2}) x^{2} + C$

Step 10.2.4

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

$x^{4} + \frac{2}{2} x^{2} + C$

Step 10.2.5

Cancel the common factor of $2$ .

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Step 10.2.5.1

Cancel the common factor.

$x^{4} + \frac{2}{2} x^{2} + C$

Step 10.2.5.2

Rewrite the expression.

$x^{4} + 1 x^{2} + C$

Step 10.2.6

Multiply $x^{2}$ by $1$ .

$x^{4} + x^{2} + C$