Calculus Examples

$\frac{d h}{d t} = \frac{17.6 t}{\sqrt{17.6 t^{2} + 1}}$

Step 1

Rewrite the equation.

$d h = \frac{17.6 t}{\sqrt{17.6 t^{2} + 1}} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d h = \int \frac{17.6 t}{\sqrt{17.6 t^{2} + 1}} d t$

Step 2.2

Apply the constant rule.

$h + C_{1} = \int \frac{17.6 t}{\sqrt{17.6 t^{2} + 1}} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $17.6$ is constant with respect to $t$ , move $17.6$ out of the integral.

$h + C_{1} = 17.6 \int \frac{t}{\sqrt{17.6 t^{2} + 1}} d t$

Step 2.3.2

Let $u = 17.6 t^{2} + 1$ . Then $d u = 35.2 t d t$ , so $\frac{1}{35.2} d u = t d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 17.6 t^{2} + 1$ . Find $\frac{d u}{d t}$ .

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Step 2.3.2.1.1

Differentiate $17.6 t^{2} + 1$ .

$\frac{d}{d t} [17.6 t^{2} + 1]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $17.6 t^{2} + 1$ with respect to $t$ is $\frac{d}{d t} [17.6 t^{2}] + \frac{d}{d t} [1]$ .

$\frac{d}{d t} [17.6 t^{2}] + \frac{d}{d t} [1]$

Step 2.3.2.1.3

Evaluate $\frac{d}{d t} [17.6 t^{2}]$ .

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Step 2.3.2.1.3.1

Since $17.6$ is constant with respect to $t$ , the derivative of $17.6 t^{2}$ with respect to $t$ is $17.6 \frac{d}{d t} [t^{2}]$ .

$17.6 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [1]$

Step 2.3.2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$17.6 (2 t) + \frac{d}{d t} [1]$

Step 2.3.2.1.3.3

Multiply $2$ by $17.6$ .

$35.2 t + \frac{d}{d t} [1]$

Step 2.3.2.1.4

Differentiate using the Constant Rule.

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Step 2.3.2.1.4.1

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

$35.2 t + 0$

Step 2.3.2.1.4.2

Add $35.2 t$ and $0$ .

$35.2 t$

Step 2.3.2.2

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

$h + C_{1} = 17.6 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{35.2} d u$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{35.2}$ .

$h + C_{1} = 17.6 \int \frac{1}{\sqrt{u} \cdot 35.2} d u$

Step 2.3.3.2

Move $35.2$ to the left of $\sqrt{u}$ .

$h + C_{1} = 17.6 \int \frac{1}{35.2 \sqrt{u}} d u$

Step 2.3.4

Since $\frac{1}{35.2}$ is constant with respect to $u$ , move $\frac{1}{35.2}$ out of the integral.

$h + C_{1} = 17.6 (\frac{1}{35.2} \int \frac{1}{\sqrt{u}} d u)$

Step 2.3.5

Combine fractions.

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Step 2.3.5.1

Combine $\frac{1}{35.2}$ and $17.6$ .

$h + C_{1} = \frac{17.6}{35.2} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.2

Apply basic rules of exponents.

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Step 2.3.5.2.1

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

$h + C_{1} = \frac{17.6}{35.2} \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 2.3.5.2.2

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

$h + C_{1} = \frac{17.6}{35.2} \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 2.3.5.2.3

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

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Step 2.3.5.2.3.1

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

$h + C_{1} = \frac{17.6}{35.2} \int u^{\frac{1}{2} \cdot - 1} d u$

Step 2.3.5.2.3.2

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

$h + C_{1} = \frac{17.6}{35.2} \int u^{\frac{- 1}{2}} d u$

Step 2.3.5.2.3.3

Move the negative in front of the fraction.

$h + C_{1} = \frac{17.6}{35.2} \int u^{- \frac{1}{2}} d u$

Step 2.3.6

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

$h + C_{1} = \frac{17.6}{35.2} (2 u^{\frac{1}{2}} + C_{2})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Rewrite $\frac{17.6}{35.2} (2 u^{\frac{1}{2}} + C_{2})$ as $0.5 (2 u^{\frac{1}{2}}) + C_{2}$ .

$h + C_{1} = 0.5 (2 u^{\frac{1}{2}}) + C_{2}$

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Multiply $2$ by $0.5$ .

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

Step 2.3.7.2.2

Multiply $u^{\frac{1}{2}}$ by $1$ .

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

Step 2.3.8

Replace all occurrences of $u$ with $17.6 t^{2} + 1$ .

$h + C_{1} = {(17.6 t^{2} + 1)}^{\frac{1}{2}} + C_{2}$

Step 2.4

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

$h = {(17.6 t^{2} + 1)}^{\frac{1}{2}} + K$