Calculus Examples

$3 e^{- 3 x} {(7 + 5 e^{- 3 x})}^{2} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int e^{- 3 x} {(7 + 5 e^{- 3 x})}^{2} d x$

Step 2

Let $u_{2} = 7 + 5 e^{- 3 x}$ . Then $d u_{2} = - 15 e^{- 3 x} d x$ , so $- \frac{1}{15} d u_{2} = e^{- 3 x} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

Let $u_{2} = 7 + 5 e^{- 3 x}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.1.1

Differentiate $7 + 5 e^{- 3 x}$ .

$\frac{d}{d x} [7 + 5 e^{- 3 x}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $7 + 5 e^{- 3 x}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [5 e^{- 3 x}]$ .

$\frac{d}{d x} [7] + \frac{d}{d x} [5 e^{- 3 x}]$

Step 2.1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [5 e^{- 3 x}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [5 e^{- 3 x}]$ .

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Step 2.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 e^{- 3 x}$ with respect to $x$ is $5 \frac{d}{d x} [e^{- 3 x}]$ .

$0 + 5 \frac{d}{d x} [e^{- 3 x}]$

Step 2.1.3.2

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 $g (x) = - 3 x$ .

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Step 2.1.3.2.1

To apply the Chain Rule, set $u_{1}$ as $- 3 x$ .

$0 + 5 (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- 3 x])$

Step 2.1.3.2.2

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

$0 + 5 (e^{u_{1}} \frac{d}{d x} [- 3 x])$

Step 2.1.3.2.3

Replace all occurrences of $u_{1}$ with $- 3 x$ .

$0 + 5 (e^{- 3 x} \frac{d}{d x} [- 3 x])$

Step 2.1.3.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$0 + 5 (e^{- 3 x} (- 3 \frac{d}{d x} [x]))$

Step 2.1.3.4

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

$0 + 5 (e^{- 3 x} (- 3 \cdot 1))$

Step 2.1.3.5

Multiply $- 3$ by $1$ .

$0 + 5 (e^{- 3 x} \cdot - 3)$

Step 2.1.3.6

Move $- 3$ to the left of $e^{- 3 x}$ .

$0 + 5 (- 3 \cdot e^{- 3 x})$

Step 2.1.3.7

Multiply $- 3$ by $5$ .

$0 - 15 e^{- 3 x}$

Step 2.1.4

Subtract $15 e^{- 3 x}$ from $0$ .

$- 15 e^{- 3 x}$

Step 2.2

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

$3 \int {u_{2}}^{2} \frac{1}{- 15} d u_{2}$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$3 \int {u_{2}}^{2} (- \frac{1}{15}) d u_{2}$

Step 3.2

Combine ${u_{2}}^{2}$ and $\frac{1}{15}$ .

$3 \int - \frac{{u_{2}}^{2}}{15} d u_{2}$

Step 4

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

$3 (- \int \frac{{u_{2}}^{2}}{15} d u_{2})$

Step 5

Multiply $- 1$ by $3$ .

$- 3 \int \frac{{u_{2}}^{2}}{15} d u_{2}$

Step 6

Since $\frac{1}{15}$ is constant with respect to $u_{2}$ , move $\frac{1}{15}$ out of the integral.

$- 3 (\frac{1}{15} \int {u_{2}}^{2} d u_{2})$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{15}$ and $- 3$ .

$\frac{- 3}{15} \int {u_{2}}^{2} d u_{2}$

Step 7.2

Cancel the common factor of $- 3$ and $15$ .

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Step 7.2.1

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{15} \int {u_{2}}^{2} d u_{2}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $3$ out of $15$ .

$\frac{3 \cdot - 1}{3 \cdot 5} \int {u_{2}}^{2} d u_{2}$

Step 7.2.2.2

Cancel the common factor.

$\frac{3 \cdot - 1}{3 \cdot 5} \int {u_{2}}^{2} d u_{2}$

Step 7.2.2.3

Rewrite the expression.

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

Step 7.3

Move the negative in front of the fraction.

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

Step 8

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

$- \frac{1}{5} (\frac{1}{3} {u_{2}}^{3} + C)$

Step 9

Simplify.

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Step 9.1

Rewrite $- \frac{1}{5} (\frac{1}{3} {u_{2}}^{3} + C)$ as $- \frac{1}{5} \cdot \frac{1}{3} {u_{2}}^{3} + C$ .

$- \frac{1}{5} \cdot \frac{1}{3} {u_{2}}^{3} + C$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{5}$ .

$- \frac{1}{3 \cdot 5} {u_{2}}^{3} + C$

Step 9.2.2

Multiply $3$ by $5$ .

$- \frac{1}{15} {u_{2}}^{3} + C$

Step 10

Replace all occurrences of $u_{2}$ with $7 + 5 e^{- 3 x}$ .

$- \frac{1}{15} {(7 + 5 e^{- 3 x})}^{3} + C$