Calculus Examples

$\int (5^{5 x + 1} - 15 x^{4}) d x$

Step 1

Remove parentheses.

$\int 5^{5 x + 1} - 15 x^{4} d x$

Step 2

Split the single integral into multiple integrals.

$\int 5^{5 x + 1} d x + \int - 15 x^{4} d x$

Step 3

Let $u = 5 x + 1$ . 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 3.1

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

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Step 3.1.1

Differentiate $5 x + 1$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

Multiply $5$ by $1$ .

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

Step 3.1.4

Differentiate using the Constant Rule.

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Step 3.1.4.1

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

$5 + 0$

Step 3.1.4.2

Add $5$ and $0$ .

$5$

Step 3.2

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

$\int 5^{u} \cdot \frac{1}{5} d u + \int - 15 x^{4} d x$

Step 4

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

$\int \frac{5^{u}}{5} d u + \int - 15 x^{4} d x$

Step 5

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

$\frac{1}{5} \int 5^{u} d u + \int - 15 x^{4} d x$

Step 6

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

$\frac{1}{5} (\frac{5^{u}}{\ln (5)} + C) + \int - 15 x^{4} d x$

Step 7

Since $- 15$ is constant with respect to $x$ , move $- 15$ out of the integral.

$\frac{1}{5} (\frac{5^{u}}{\ln (5)} + C) - 15 \int x^{4} d x$

Step 8

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{5} (\frac{5^{u}}{\ln (5)} + C) - 15 (\frac{1}{5} x^{5} + C)$

Step 9

Simplify.

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Step 9.1

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

$\frac{1}{5} (\frac{5^{u}}{\ln (5)} + C) - 15 (\frac{x^{5}}{5} + C)$

Step 9.2

Simplify.

$\frac{1 \cdot 5^{u}}{5 \ln (5)} - 15 \frac{x^{5}}{5} + C$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $5^{u}$ by $1$ .

$\frac{5^{u}}{5 \ln (5)} - 15 \frac{x^{5}}{5} + C$

Step 9.3.2

Cancel the common factor of $5^{u}$ and $5$ .

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Step 9.3.2.1

Factor $5$ out of $5^{u}$ .

$\frac{5 \cdot 5^{u - 1}}{5 \ln (5)} - 15 \frac{x^{5}}{5} + C$

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $5$ out of $5 \ln (5)$ .

$\frac{5 \cdot 5^{u - 1}}{5 (\ln (5))} - 15 \frac{x^{5}}{5} + C$

Step 9.3.2.2.2

Cancel the common factor.

$\frac{5 \cdot 5^{u - 1}}{5 \ln (5)} - 15 \frac{x^{5}}{5} + C$

Step 9.3.2.2.3

Rewrite the expression.

$\frac{5^{u - 1}}{\ln (5)} - 15 \frac{x^{5}}{5} + C$

Step 9.3.3

Combine $- 15$ and $\frac{x^{5}}{5}$ .

$\frac{5^{u - 1}}{\ln (5)} + \frac{- 15 x^{5}}{5} + C$

Step 9.3.4

Cancel the common factor of $- 15$ and $5$ .

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Step 9.3.4.1

Factor $5$ out of $- 15 x^{5}$ .

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

Step 9.3.4.2

Cancel the common factors.

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Step 9.3.4.2.1

Factor $5$ out of $5$ .

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

Step 9.3.4.2.2

Cancel the common factor.

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

Step 9.3.4.2.3

Rewrite the expression.

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

Step 9.3.4.2.4

Divide $- 3 x^{5}$ by $1$ .

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

Step 9.3.5

To write $- 3 x^{5}$ as a fraction with a common denominator, multiply by $\frac{\ln (5)}{\ln (5)}$ .

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

Step 9.3.6

Combine $- 3 x^{5}$ and $\frac{\ln (5)}{\ln (5)}$ .

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

Step 9.3.7

Combine the numerators over the common denominator.

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

Step 10

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

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

Step 11

Reorder terms.

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