Calculus Examples

$\frac{d y}{d x} = \frac{4}{\sqrt{x}}$

Step 1

Rewrite the equation.

$d y = \frac{4}{\sqrt{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{4}{\sqrt{x}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{4}{\sqrt{x}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$y + C_{1} = 4 \int \frac{1}{\sqrt{x}} d x$

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

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

$y + C_{1} = 4 \int \frac{1}{x^{\frac{1}{2}}} d x$

Step 2.3.2.2

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

$y + C_{1} = 4 \int {(x^{\frac{1}{2}})}^{- 1} d x$

Step 2.3.2.3

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

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Step 2.3.2.3.1

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

$y + C_{1} = 4 \int x^{\frac{1}{2} \cdot - 1} d x$

Step 2.3.2.3.2

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

$y + C_{1} = 4 \int x^{\frac{- 1}{2}} d x$

Step 2.3.2.3.3

Move the negative in front of the fraction.

$y + C_{1} = 4 \int x^{- \frac{1}{2}} d x$

Step 2.3.3

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

$y + C_{1} = 4 (2 x^{\frac{1}{2}} + C_{2})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $4 (2 x^{\frac{1}{2}} + C_{2})$ as $4 \cdot 2 x^{\frac{1}{2}} + C_{2}$ .

$y + C_{1} = 4 \cdot 2 x^{\frac{1}{2}} + C_{2}$

Step 2.3.4.2

Multiply $4$ by $2$ .

$y + C_{1} = 8 x^{\frac{1}{2}} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = 8 x^{\frac{1}{2}} + K$