Calculus Examples

$\int (\sqrt{t^{6} + 5 t}) (6 t^{5} + 5) d t$

Step 1

Let $u = t^{6} + 5 t$ . Then $d u = (6 t^{5} + 5) d t$ , so $\frac{1}{6 t^{5} + 5} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = t^{6} + 5 t$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $t^{6} + 5 t$ .

$\frac{d}{d t} [t^{6} + 5 t]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $t^{6} + 5 t$ with respect to $t$ is $\frac{d}{d t} [t^{6}] + \frac{d}{d t} [5 t]$ .

$\frac{d}{d t} [t^{6}] + \frac{d}{d t} [5 t]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 6$ .

$6 t^{5} + \frac{d}{d t} [5 t]$

Step 1.1.3

Evaluate $\frac{d}{d t} [5 t]$ .

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Step 1.1.3.1

Since $5$ is constant with respect to $t$ , the derivative of $5 t$ with respect to $t$ is $5 \frac{d}{d t} [t]$ .

$6 t^{5} + 5 \frac{d}{d t} [t]$

Step 1.1.3.2

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

$6 t^{5} + 5 \cdot 1$

Step 1.1.3.3

Multiply $5$ by $1$ .

$6 t^{5} + 5$

Step 1.2

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

$\int \sqrt{u} d u$

Step 2

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

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

Step 3

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

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

Step 4

Replace all occurrences of $u$ with $t^{6} + 5 t$ .

$\frac{2}{3} {(t^{6} + 5 t)}^{\frac{3}{2}} + C$