Calculus Examples

\int x \sqrt{x} d x

$\int x \sqrt{x} d x$

Step 1

Simplify.

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Step 1.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

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

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

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

Step 1.2

Multiply

x

$x$ by

x^{\frac{1}{2}}

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

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Step 1.2.1

Multiply

x

$x$ by

x^{\frac{1}{2}}

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

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Step 1.2.1.1

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 1.2.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 1.2.2

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 1.2.3

Combine the numerators over the common denominator.

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

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

Step 1.2.4

Add

2

$2$ and

1

$1$ .

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

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

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

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

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

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

Step 2

By the Power Rule, the integral of

x^{\frac{3}{2}}

$x^{\frac{3}{2}}$ with respect to

x

$x$ is

\frac{2}{5} x^{\frac{5}{2}}

$\frac{2}{5} x^{\frac{5}{2}}$ .

\frac{2}{5} x^{\frac{5}{2}} + C

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

$\int_{}^{} x \sqrt[]{x}$

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⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$