Calculus Examples

$\int x^{3} e^{6 x^{4}} \sqrt{{(2 e^{6 x^{4}} + 1)}^{5}} d x$

Step 1

Let $u_{2} = 2 e^{6 x^{4}} + 1$ . Then $d u_{2} = 48 x^{3} e^{6 x^{4}} d x$ , so $\frac{1}{48} d u_{2} = x^{3} e^{6 x^{4}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = 2 e^{6 x^{4}} + 1$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $2 e^{6 x^{4}} + 1$ .

$\frac{d}{d x} [2 e^{6 x^{4}} + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $2 e^{6 x^{4}} + 1$ with respect to $x$ is $\frac{d}{d x} [2 e^{6 x^{4}}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [2 e^{6 x^{4}}] + \frac{d}{d x} [1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 e^{6 x^{4}}]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{6 x^{4}}$ with respect to $x$ is $2 \frac{d}{d x} [e^{6 x^{4}}]$ .

$2 \frac{d}{d x} [e^{6 x^{4}}] + \frac{d}{d x} [1]$

Step 1.1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 6 x^{4}$ .

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Step 1.1.3.2.1

To apply the Chain Rule, set $u_{1}$ as $6 x^{4}$ .

$2 (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [6 x^{4}]) + \frac{d}{d x} [1]$

Step 1.1.3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$2 (e^{u_{1}} \frac{d}{d x} [6 x^{4}]) + \frac{d}{d x} [1]$

Step 1.1.3.2.3

Replace all occurrences of $u_{1}$ with $6 x^{4}$ .

$2 (e^{6 x^{4}} \frac{d}{d x} [6 x^{4}]) + \frac{d}{d x} [1]$

Step 1.1.3.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4}$ with respect to $x$ is $6 \frac{d}{d x} [x^{4}]$ .

$2 (e^{6 x^{4}} (6 \frac{d}{d x} [x^{4}])) + \frac{d}{d x} [1]$

Step 1.1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$2 (e^{6 x^{4}} (6 (4 x^{3}))) + \frac{d}{d x} [1]$

Step 1.1.3.5

Multiply $4$ by $6$ .

$2 (e^{6 x^{4}} (24 x^{3})) + \frac{d}{d x} [1]$

Step 1.1.3.6

Multiply $24$ by $2$ .

$48 e^{6 x^{4}} x^{3} + \frac{d}{d x} [1]$

Step 1.1.4

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

$48 e^{6 x^{4}} x^{3} + 0$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Add $48 e^{6 x^{4}} x^{3}$ and $0$ .

$48 e^{6 x^{4}} x^{3}$

Step 1.1.5.2

Reorder factors in $48 e^{6 x^{4}} x^{3}$ .

$48 x^{3} e^{6 x^{4}}$

Step 1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int \sqrt{{u_{2}}^{5}} \frac{1}{48} d u_{2}$

Step 2

Combine $\sqrt{{u_{2}}^{5}}$ and $\frac{1}{48}$ .

$\int \frac{\sqrt{{u_{2}}^{5}}}{48} d u_{2}$

Step 3

Since $\frac{1}{48}$ is constant with respect to $u_{2}$ , move $\frac{1}{48}$ out of the integral.

$\frac{1}{48} \int \sqrt{{u_{2}}^{5}} d u_{2}$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{u_{2}}^{5}}$ as ${u_{2}}^{\frac{5}{2}}$ .

$\frac{1}{48} \int {u_{2}}^{\frac{5}{2}} d u_{2}$

Step 5

By the Power Rule, the integral of ${u_{2}}^{\frac{5}{2}}$ with respect to $u_{2}$ is $\frac{2}{7} {u_{2}}^{\frac{7}{2}}$ .

$\frac{1}{48} (\frac{2}{7} {u_{2}}^{\frac{7}{2}} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{48} (\frac{2}{7} {u_{2}}^{\frac{7}{2}} + C)$ as $\frac{1}{48} \cdot \frac{2}{7} {u_{2}}^{\frac{7}{2}} + C$ .

$\frac{1}{48} \cdot \frac{2}{7} {u_{2}}^{\frac{7}{2}} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{48}$ by $\frac{2}{7}$ .

$\frac{2}{48 \cdot 7} {u_{2}}^{\frac{7}{2}} + C$

Step 6.2.2

Multiply $48$ by $7$ .

$\frac{2}{336} {u_{2}}^{\frac{7}{2}} + C$

Step 6.2.3

Cancel the common factor of $2$ and $336$ .

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Step 6.2.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{336} {u_{2}}^{\frac{7}{2}} + C$

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $2$ out of $336$ .

$\frac{2 \cdot 1}{2 \cdot 168} {u_{2}}^{\frac{7}{2}} + C$

Step 6.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 168} {u_{2}}^{\frac{7}{2}} + C$

Step 6.2.3.2.3

Rewrite the expression.

$\frac{1}{168} {u_{2}}^{\frac{7}{2}} + C$

Step 7

Replace all occurrences of $u_{2}$ with $2 e^{6 x^{4}} + 1$ .

$\frac{1}{168} {(2 e^{6 x^{4}} + 1)}^{\frac{7}{2}} + C$