Calculus Examples

$\int \frac{5}{e^{5 x}} - \frac{2}{5 x} d x$

Step 1

Split the single integral into multiple integrals.

$\int \frac{5}{e^{5 x}} d x + \int - \frac{2}{5 x} d x$

Step 2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \frac{1}{e^{5 x}} d x + \int - \frac{2}{5 x} d x$

Step 3

Simplify the expression.

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Step 3.1

Negate the exponent of $e^{5 x}$ and move it out of the denominator.

$5 \int 1 {(e^{5 x})}^{- 1} d x + \int - \frac{2}{5 x} d x$

Step 3.2

Simplify.

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Step 3.2.1

Multiply the exponents in ${(e^{5 x})}^{- 1}$ .

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Step 3.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 \int 1 e^{5 x \cdot - 1} d x + \int - \frac{2}{5 x} d x$

Step 3.2.1.2

Multiply $- 1$ by $5$ .

$5 \int 1 e^{- 5 x} d x + \int - \frac{2}{5 x} d x$

Step 3.2.2

Multiply $e^{- 5 x}$ by $1$ .

$5 \int e^{- 5 x} d x + \int - \frac{2}{5 x} d x$

Step 4

Let $u = - 5 x$ . 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 4.1

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

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Step 4.1.1

Differentiate $- 5 x$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

$- 5 \cdot 1$

Step 4.1.4

Multiply $- 5$ by $1$ .

$- 5$

Step 4.2

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

$5 \int e^{u} \frac{1}{- 5} d u + \int - \frac{2}{5 x} d x$

Step 5

Simplify.

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Step 5.1

Move the negative in front of the fraction.

$5 \int e^{u} (- \frac{1}{5}) d u + \int - \frac{2}{5 x} d x$

Step 5.2

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

$5 \int - \frac{e^{u}}{5} d u + \int - \frac{2}{5 x} d x$

Step 6

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

$5 (- \int \frac{e^{u}}{5} d u) + \int - \frac{2}{5 x} d x$

Step 7

Multiply $- 1$ by $5$ .

$- 5 \int \frac{e^{u}}{5} d u + \int - \frac{2}{5 x} d x$

Step 8

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

$- 5 (\frac{1}{5} \int e^{u} d u) + \int - \frac{2}{5 x} d x$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{5}$ and $- 5$ .

$\frac{- 5}{5} \int e^{u} d u + \int - \frac{2}{5 x} d x$

Step 9.2

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

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Step 9.2.1

Factor $5$ out of $- 5$ .

$\frac{5 \cdot - 1}{5} \int e^{u} d u + \int - \frac{2}{5 x} d x$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $5$ out of $5$ .

$\frac{5 \cdot - 1}{5 (1)} \int e^{u} d u + \int - \frac{2}{5 x} d x$

Step 9.2.2.2

Cancel the common factor.

$\frac{5 \cdot - 1}{5 \cdot 1} \int e^{u} d u + \int - \frac{2}{5 x} d x$

Step 9.2.2.3

Rewrite the expression.

$\frac{- 1}{1} \int e^{u} d u + \int - \frac{2}{5 x} d x$

Step 9.2.2.4

Divide $- 1$ by $1$ .

$- \int e^{u} d u + \int - \frac{2}{5 x} d x$

Step 10

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$- (e^{u} + C) + \int - \frac{2}{5 x} d x$

Step 11

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

$- (e^{u} + C) - \int \frac{2}{5 x} d x$

Step 12

Since $\frac{2}{5}$ is constant with respect to $x$ , move $\frac{2}{5}$ out of the integral.

$- (e^{u} + C) - (\frac{2}{5} \int \frac{1}{x} d x)$

Step 13

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$- (e^{u} + C) - \frac{2}{5} (\ln (| x |) + C)$

Step 14

Simplify.

$- e^{u} - \frac{2}{5} \ln (| x |) + C$

Step 15

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

$- e^{- 5 x} - \frac{2}{5} \ln (| x |) + C$