Calculus Examples

$\int_{1}^{3} \frac{2 + y e^{y}}{y} d y$

Step 1

Split the fraction into multiple fractions.

$\int_{1}^{3} \frac{2}{y} + \frac{y e^{y}}{y} d y$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{3} \frac{2}{y} d y + \int_{1}^{3} \frac{y e^{y}}{y} d y$

Step 3

Cancel the common factor of $y$ .

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Step 3.1

Cancel the common factor.

$\int_{1}^{3} \frac{2}{y} d y + \int_{1}^{3} \frac{y e^{y}}{y} d y$

Step 3.2

Divide $e^{y}$ by $1$ .

$\int_{1}^{3} \frac{2}{y} d y + \int_{1}^{3} e^{y} d y$

Step 4

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$2 \int_{1}^{3} \frac{1}{y} d y + \int_{1}^{3} e^{y} d y$

Step 5

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

$2 (\ln (| y |)]_{1}^{3}) + \int_{1}^{3} e^{y} d y$

Step 6

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

$2 (\ln (| y |)]_{1}^{3}) + e^{y}]_{1}^{3}$

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

Evaluate $\ln (| y |)$ at $3$ and at $1$ .

$2 ((\ln (| 3 |)) - \ln (| 1 |)) + e^{y}]_{1}^{3}$

Step 7.1.2

Evaluate $e^{y}$ at $3$ and at $1$ .

$2 (\ln (| 3 |) - \ln (| 1 |)) + e^{3} - e^{1}$

Step 7.1.3

Simplify.

$2 (\ln (| 3 |) - \ln (| 1 |)) + e^{3} - e$

Step 7.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$2 \ln (\frac{| 3 |}{| 1 |}) + e^{3} - e$

Step 7.3

Simplify.

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Step 7.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$2 \ln (\frac{3}{| 1 |}) + e^{3} - e$

Step 7.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$2 \ln (\frac{3}{1}) + e^{3} - e$

Step 7.3.3

Divide $3$ by $1$ .

$2 \ln (3) + e^{3} - e$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 \ln (3) + e^{3} - e$

Decimal Form:

$19.56447967 \dots$

Step 9