Calculus Examples

$\int 12 x - \frac{4}{\sqrt{x}} d x$

Step 1

Split the single integral into multiple integrals.

$\int 12 x d x + \int - \frac{4}{\sqrt{x}} d x$

Step 2

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int x d x + \int - \frac{4}{\sqrt{x}} d x$

Step 3

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

$12 (\frac{1}{2} x^{2} + C) + \int - \frac{4}{\sqrt{x}} d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$12 (\frac{1}{2} x^{2} + C) - \int \frac{4}{\sqrt{x}} d x$

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$12 (\frac{1}{2} x^{2} + C) - (4 \int \frac{1}{\sqrt{x}} d x)$

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

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

$12 (\frac{x^{2}}{2} + C) - (4 \int \frac{1}{\sqrt{x}} d x)$

Step 6.1.2

Multiply $4$ by $- 1$ .

$12 (\frac{x^{2}}{2} + C) - 4 \int \frac{1}{\sqrt{x}} d x$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

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

$12 (\frac{x^{2}}{2} + C) - 4 \int \frac{1}{x^{\frac{1}{2}}} d x$

Step 6.2.2

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

$12 (\frac{x^{2}}{2} + C) - 4 \int {(x^{\frac{1}{2}})}^{- 1} d x$

Step 6.2.3

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

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

$12 (\frac{x^{2}}{2} + C) - 4 \int x^{\frac{- 1}{2}} d x$

Step 6.2.3.3

Move the negative in front of the fraction.

$12 (\frac{x^{2}}{2} + C) - 4 \int x^{- \frac{1}{2}} d x$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

$6 x^{2} - 4 (2 x^{\frac{1}{2}}) + C$

Step 8.2

Multiply $2$ by $- 4$ .

$6 x^{2} - 8 x^{\frac{1}{2}} + C$