Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 2

Simplify.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 2.2

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.2.3

Combine the numerators over the common denominator.

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

Step 2.2.4

Add $2$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

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

Step 5

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

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

Step 6

Apply basic rules of exponents.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Multiply $- 1$ by $2$ .

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