Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 1.2.1

Multiply $x$ by $x^{3}$ .

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.2

Add $1$ and $3$ .

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Factor $x$ out of $x^{3}$ .

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

Step 1.3.2

Cancel the common factor.

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

Step 1.3.3

Rewrite the expression.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{5} x^{5} + C + \int \frac{2}{x^{2}} d x$

Step 4

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

$\frac{1}{5} x^{5} + C + 2 \int \frac{1}{x^{2}} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

$\frac{1}{5} x^{5} + C + 2 \int x^{2 \cdot - 1} d x$

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

$\frac{1}{5} x^{5} + C + 2 (- x^{- 1} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$\frac{1}{5} x^{5} + 2 (- x^{- 1}) + C$

Step 7.2

Multiply $- 1$ by $2$ .

$\frac{1}{5} x^{5} - 2 x^{- 1} + C$