Calculus Examples

$\int \frac{6}{\sqrt{x}} + 6 \sqrt{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int \frac{6}{\sqrt{x}} d x + \int 6 \sqrt{x} d x$

Step 2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \frac{1}{\sqrt{x}} d x + \int 6 \sqrt{x} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

$6 \int \frac{1}{x^{\frac{1}{2}}} d x + \int 6 \sqrt{x} d x$

Step 3.2

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

$6 \int {(x^{\frac{1}{2}})}^{- 1} d x + \int 6 \sqrt{x} d x$

Step 3.3

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

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Step 3.3.1

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

$6 \int x^{\frac{1}{2} \cdot - 1} d x + \int 6 \sqrt{x} d x$

Step 3.3.2

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

$6 \int x^{\frac{- 1}{2}} d x + \int 6 \sqrt{x} d x$

Step 3.3.3

Move the negative in front of the fraction.

$6 \int x^{- \frac{1}{2}} d x + \int 6 \sqrt{x} d x$

Step 4

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

$6 (2 x^{\frac{1}{2}} + C) + \int 6 \sqrt{x} d x$

Step 5

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 (2 x^{\frac{1}{2}} + C) + 6 \int \sqrt{x} d x$

Step 6

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

$6 (2 x^{\frac{1}{2}} + C) + 6 \int x^{\frac{1}{2}} d x$

Step 7

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

$6 (2 x^{\frac{1}{2}} + C) + 6 (\frac{2}{3} x^{\frac{3}{2}} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$12 x^{\frac{1}{2}} + 6 (\frac{2}{3} x^{\frac{3}{2}}) + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$12 x^{\frac{1}{2}} + 6 \frac{2 x^{\frac{3}{2}}}{3} + C$

Step 8.2.2

Combine $6$ and $\frac{2 x^{\frac{3}{2}}}{3}$ .

$12 x^{\frac{1}{2}} + \frac{6 (2 x^{\frac{3}{2}})}{3} + C$

Step 8.2.3

Multiply $2$ by $6$ .

$12 x^{\frac{1}{2}} + \frac{12 x^{\frac{3}{2}}}{3} + C$

Step 8.2.4

Factor $3$ out of $12 x^{\frac{3}{2}}$ .

$12 x^{\frac{1}{2}} + \frac{3 (4 x^{\frac{3}{2}})}{3} + C$

Step 8.2.5

Cancel the common factors.

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Step 8.2.5.1

Factor $3$ out of $3$ .

$12 x^{\frac{1}{2}} + \frac{3 (4 x^{\frac{3}{2}})}{3 (1)} + C$

Step 8.2.5.2

Cancel the common factor.

$12 x^{\frac{1}{2}} + \frac{3 (4 x^{\frac{3}{2}})}{3 \cdot 1} + C$

Step 8.2.5.3

Rewrite the expression.

$12 x^{\frac{1}{2}} + \frac{4 x^{\frac{3}{2}}}{1} + C$

Step 8.2.5.4

Divide $4 x^{\frac{3}{2}}$ by $1$ .

$12 x^{\frac{1}{2}} + 4 x^{\frac{3}{2}} + C$