Calculus Examples

$\int 5 (5^{5 x}) d x$

Step 1

Multiply $5$ by $5^{5 x}$ .

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Step 1.1

Raise $5$ to the power of $1$ .

$\int 5^{1} \cdot 5^{5 x} d x$

Step 1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 5^{1 + 5 x} d x$

Step 2

Let $u = 1 + 5 x$ . 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 2.1

Let $u = 1 + 5 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $1 + 5 x$ .

$\frac{d}{d x} [1 + 5 x]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $1 + 5 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [5 x]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [5 x]$

Step 2.1.2.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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 2.1.3.2

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 2.1.3.3

Multiply $5$ by $1$ .

$0 + 5$

Step 2.1.4

Add $0$ and $5$ .

$5$

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

The integral of $5^{u}$ with respect to $u$ is $\frac{5^{u}}{\ln (5)}$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Replace all occurrences of $u$ with $1 + 5 x$ .

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