Calculus Examples

$\int 3^{2 - 5 x} d x$

Step 1

Let $u = 2 - 5 x$ . Then $d u = - 5 d x$ , so $- \frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 - 5 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 - 5 x$ .

$\frac{d}{d x} [2 - 5 x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $2 - 5 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [- 5 x]$

Step 1.1.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

$0 - 5 \frac{d}{d x} [x]$

Step 1.1.3.2

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

$0 - 5 \cdot 1$

Step 1.1.3.3

Multiply $- 5$ by $1$ .

$0 - 5$

Step 1.1.4

Subtract $5$ from $0$ .

$- 5$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int 3^{u} \cdot (- \frac{1}{5}) d u$

Step 2.2

Combine $3^{u}$ and $\frac{1}{5}$ .

$\int - \frac{3^{u}}{5} d u$

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \frac{3^{u}}{5} d u$

Step 4

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

$- (\frac{1}{5} \int 3^{u} d u)$

Step 5

The integral of $3^{u}$ with respect to $u$ is $\frac{3^{u}}{\ln (3)}$ .

$- \frac{1}{5} (\frac{3^{u}}{\ln (3)} + C)$

Step 6

Simplify.

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Step 6.1

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

$- \frac{1}{5} \cdot \frac{1}{\ln (3)} \cdot 3^{u} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{\ln (3)}$ by $\frac{1}{5}$ .

$- \frac{1}{\ln (3) \cdot 5} \cdot 3^{u} + C$

Step 6.2.2

Move $5$ to the left of $\ln (3)$ .

$- \frac{1}{5 \ln (3)} \cdot 3^{u} + C$

Step 7

Replace all occurrences of $u$ with $2 - 5 x$ .

$- \frac{1}{5 \ln (3)} \cdot 3^{2 - 5 x} + C$