Calculus Examples

$\int \frac{1}{\sqrt{x^{3}}} + 5 e^{5 x} d x$

Step 1

Split the single integral into multiple integrals.

$\int \frac{1}{\sqrt{x^{3}}} d x + \int 5 e^{5 x} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

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

Step 2.2

Move $x^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.3

Multiply the exponents in ${(x^{\frac{3}{2}})}^{- 1}$ .

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Step 2.3.1

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

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

Step 2.3.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 2.3.2.1

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

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

Step 2.3.2.2

Multiply $3$ by $- 1$ .

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 5

Let $u = 5 x$ . Then $d u = 5 d x$ , so $\frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 5.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$5 \cdot 1$

Step 5.1.4

Multiply $5$ by $1$ .

$5$

Step 5.2

Rewrite the problem using $u$ and $d u$ .

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

Step 6

Combine $e^{u}$ and $\frac{1}{5}$ .

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

Step 7

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

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

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{5}$ and $5$ .

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

Step 8.2

Cancel the common factor of $5$ .

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Step 8.2.1

Cancel the common factor.

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

Step 8.2.2

Rewrite the expression.

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

Step 8.3

Multiply $\int e^{u} d u$ by $1$ .

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

Step 9

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify.

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Step 10.2.1

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

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

Step 10.2.2

Move the negative in front of the fraction.

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

Step 11

Replace all occurrences of $u$ with $5 x$ .

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