Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.3.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1]$

Step 2.3.1.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.1.1.5

Add $1$ and $0$ .

$1$

Step 2.3.1.2

Rewrite the problem using $u$ and $d u$ .

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

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 $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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Step 2.3.2.3.1

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

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

Step 2.3.2.3.2

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

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

Step 2.3.2.3.3

Move the negative in front of the fraction.

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

Step 2.3.3

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

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

Step 2.3.4

Replace all occurrences of $u$ with $x + 1$ .

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

Step 2.4

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

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