Calculus Examples

$\int e^{2 x} {(1 + e^{2 x})}^{3} d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$0 + \frac{d}{d x} [e^{2 x}]$

Step 1.1.3

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

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Step 1.1.3.1

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 $g (x) = 2 x$ .

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Step 1.1.3.1.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$0 + \frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [2 x]$

Step 1.1.3.1.2

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

$0 + e^{u_{1}} \frac{d}{d x} [2 x]$

Step 1.1.3.1.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$0 + e^{2 x} \frac{d}{d x} [2 x]$

Step 1.1.3.2

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

$0 + e^{2 x} (2 \frac{d}{d x} [x])$

Step 1.1.3.3

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

$0 + e^{2 x} (2 \cdot 1)$

Step 1.1.3.4

Multiply $2$ by $1$ .

$0 + e^{2 x} \cdot 2$

Step 1.1.3.5

Move $2$ to the left of $e^{2 x}$ .

$0 + 2 e^{2 x}$

Step 1.1.4

Add $0$ and $2 e^{2 x}$ .

$2 e^{2 x}$

Step 1.2

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

$\int {u_{2}}^{3} \frac{1}{2} d u_{2}$

Step 2

Combine ${u_{2}}^{3}$ and $\frac{1}{2}$ .

$\int \frac{{u_{2}}^{3}}{2} d u_{2}$

Step 3

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

$\frac{1}{2} \int {u_{2}}^{3} d u_{2}$

Step 4

By the Power Rule, the integral of ${u_{2}}^{3}$ with respect to $u_{2}$ is $\frac{1}{4} {u_{2}}^{4}$ .

$\frac{1}{2} (\frac{1}{4} {u_{2}}^{4} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{2} (\frac{1}{4} {u_{2}}^{4} + C)$ as $\frac{1}{2} \cdot \frac{1}{4} {u_{2}}^{4} + C$ .

$\frac{1}{2} \cdot \frac{1}{4} {u_{2}}^{4} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$\frac{1}{2 \cdot 4} {u_{2}}^{4} + C$

Step 5.2.2

Multiply $2$ by $4$ .

$\frac{1}{8} {u_{2}}^{4} + C$

Step 6

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

$\frac{1}{8} {(1 + e^{2 x})}^{4} + C$