Calculus Examples

$\int t \sqrt{t^{2} + 1} d t$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $t^{2} + 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$2 t + 0$

Step 1.1.5

Add $2 t$ and $0$ .

$2 t$

Step 1.2

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

$\int \sqrt{u} \frac{1}{2} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{2}$ .

$\int \frac{\sqrt{u}}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \sqrt{u} d u$

Step 4

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

$\frac{1}{2} \int u^{\frac{1}{2}} d u$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $\frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$\frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

$\frac{2}{2 \cdot 3} u^{\frac{3}{2}} + C$

Step 6.2.2

Multiply $2$ by $3$ .

$\frac{2}{6} u^{\frac{3}{2}} + C$

Step 6.2.3

Cancel the common factor of $2$ and $6$ .

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Step 6.2.3.1

Factor $2$ out of $2$ .

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

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3} u^{\frac{3}{2}} + C$

Step 6.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3} u^{\frac{3}{2}} + C$

Step 6.2.3.2.3

Rewrite the expression.

$\frac{1}{3} u^{\frac{3}{2}} + C$

Step 7

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

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