Calculus Examples

$\int \frac{28 x}{\sqrt{x}} d x$

Step 1

Since $28$ is constant with respect to $x$ , move $28$ out of the integral.

$28 \int \frac{x}{\sqrt{x}} d x$

Step 2

Simplify the expression.

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Step 2.1

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

$28 \int \frac{x}{x^{\frac{1}{2}}} d x$

Step 2.2

Simplify.

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Step 2.2.1

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$28 \int x \cdot x^{- \frac{1}{2}} d x$

Step 2.2.2

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

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

$28 \int x^{1} x^{- \frac{1}{2}} d x$

Step 2.2.2.1.2

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

$28 \int x^{1 - \frac{1}{2}} d x$

Step 2.2.2.2

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

$28 \int x^{\frac{2}{2} - \frac{1}{2}} d x$

Step 2.2.2.3

Combine the numerators over the common denominator.

$28 \int x^{\frac{2 - 1}{2}} d x$

Step 2.2.2.4

Subtract $1$ from $2$ .

$28 \int x^{\frac{1}{2}} d x$

Step 3

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

$28 (\frac{2}{3} x^{\frac{3}{2}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $28 (\frac{2}{3} x^{\frac{3}{2}} + C)$ as $28 (\frac{2}{3}) x^{\frac{3}{2}} + C$ .

$28 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $28$ and $\frac{2}{3}$ .

$\frac{28 \cdot 2}{3} x^{\frac{3}{2}} + C$

Step 4.2.2

Multiply $28$ by $2$ .

$\frac{56}{3} x^{\frac{3}{2}} + C$