Calculus Examples

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

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Add $2$ and $1$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

$10 x^{\frac{1}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C$