Calculus Examples

$\int 5 t^{4} \sqrt{t^{5} + 4} d t$

Step 1

Since $5$ is constant with respect to $t$ , move $5$ out of the integral.

$5 \int t^{4} \sqrt{t^{5} + 4} d t$

Step 2

Let $u = t^{5} + 4$ . Then $d u = 5 t^{4} d t$ , so $\frac{1}{5} d u = t^{4} d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = t^{5} + 4$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $t^{5} + 4$ .

$\frac{d}{d t} [t^{5} + 4]$

Step 2.1.2

By the Sum Rule, the derivative of $t^{5} + 4$ with respect to $t$ is $\frac{d}{d t} [t^{5}] + \frac{d}{d t} [4]$ .

$\frac{d}{d t} [t^{5}] + \frac{d}{d t} [4]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 5$ .

$5 t^{4} + \frac{d}{d t} [4]$

Step 2.1.4

Since $4$ is constant with respect to $t$ , the derivative of $4$ with respect to $t$ is $0$ .

$5 t^{4} + 0$

Step 2.1.5

Add $5 t^{4}$ and $0$ .

$5 t^{4}$

Step 2.2

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

$5 \int \sqrt{u} \frac{1}{5} d u$

Step 3

Combine $\sqrt{u}$ and $\frac{1}{5}$ .

$5 \int \frac{\sqrt{u}}{5} d u$

Step 4

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$5 (\frac{1}{5} \int \sqrt{u} d u)$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $\frac{1}{5}$ and $5$ .

$\frac{5}{5} \int \sqrt{u} d u$

Step 5.1.2

Cancel the common factor of $5$ .

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Step 5.1.2.1

Cancel the common factor.

$\frac{5}{5} \int \sqrt{u} d u$

Step 5.1.2.2

Rewrite the expression.

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

Step 5.1.3

Multiply $\int \sqrt{u} d u$ by $1$ .

$\int \sqrt{u} d u$

Step 5.2

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 6

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 7

Replace all occurrences of $u$ with $t^{5} + 4$ .

$\frac{2}{3} {(t^{5} + 4)}^{\frac{3}{2}} + C$