Calculus Examples

$\int \frac{1}{3 y} - \frac{5}{\sqrt{y}} + e^{\frac{- y}{2}} d y$

Step 1

Move the negative in front of the fraction.

$\int \frac{1}{3 y} - \frac{5}{\sqrt{y}} + e^{- \frac{y}{2}} d y$

Step 2

Split the single integral into multiple integrals.

$\int \frac{1}{3 y} d y + \int - \frac{5}{\sqrt{y}} d y + \int e^{- \frac{y}{2}} d y$

Step 3

Since $\frac{1}{3}$ is constant with respect to $y$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \frac{1}{y} d y + \int - \frac{5}{\sqrt{y}} d y + \int e^{- \frac{y}{2}} d y$

Step 4

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

$\frac{1}{3} (\ln (| y |) + C) + \int - \frac{5}{\sqrt{y}} d y + \int e^{- \frac{y}{2}} d y$

Step 5

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

$\frac{1}{3} (\ln (| y |) + C) - \int \frac{5}{\sqrt{y}} d y + \int e^{- \frac{y}{2}} d y$

Step 6

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

$\frac{1}{3} (\ln (| y |) + C) - (5 \int \frac{1}{\sqrt{y}} d y) + \int e^{- \frac{y}{2}} d y$

Step 7

Simplify the expression.

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Step 7.1

Multiply $5$ by $- 1$ .

$\frac{1}{3} (\ln (| y |) + C) - 5 \int \frac{1}{\sqrt{y}} d y + \int e^{- \frac{y}{2}} d y$

Step 7.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\frac{1}{3} (\ln (| y |) + C) - 5 \int \frac{1}{y^{\frac{1}{2}}} d y + \int e^{- \frac{y}{2}} d y$

Step 7.3

Move $y^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{3} (\ln (| y |) + C) - 5 \int {(y^{\frac{1}{2}})}^{- 1} d y + \int e^{- \frac{y}{2}} d y$

Step 7.4

Multiply the exponents in ${(y^{\frac{1}{2}})}^{- 1}$ .

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Step 7.4.1

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

$\frac{1}{3} (\ln (| y |) + C) - 5 \int y^{\frac{1}{2} \cdot - 1} d y + \int e^{- \frac{y}{2}} d y$

Step 7.4.2

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{1}{3} (\ln (| y |) + C) - 5 \int y^{\frac{- 1}{2}} d y + \int e^{- \frac{y}{2}} d y$

Step 7.4.3

Move the negative in front of the fraction.

$\frac{1}{3} (\ln (| y |) + C) - 5 \int y^{- \frac{1}{2}} d y + \int e^{- \frac{y}{2}} d y$

Step 8

By the Power Rule, the integral of $y^{- \frac{1}{2}}$ with respect to $y$ is $2 y^{\frac{1}{2}}$ .

$\frac{1}{3} (\ln (| y |) + C) - 5 (2 y^{\frac{1}{2}} + C) + \int e^{- \frac{y}{2}} d y$

Step 9

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

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Step 9.1

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

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Step 9.1.1

Differentiate $- \frac{y}{2}$ .

$\frac{d}{d y} [- \frac{y}{2}]$

Step 9.1.2

Since $- \frac{1}{2}$ is constant with respect to $y$ , the derivative of $- \frac{y}{2}$ with respect to $y$ is $- \frac{1}{2} \frac{d}{d y} [y]$ .

$- \frac{1}{2} \frac{d}{d y} [y]$

Step 9.1.3

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

$- \frac{1}{2} \cdot 1$

Step 9.1.4

Multiply $- 1$ by $1$ .

$- \frac{1}{2}$

Step 9.2

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

$\frac{1}{3} (\ln (| y |) + C) - 5 (2 y^{\frac{1}{2}} + C) + \int - e^{u} \frac{- 1}{- \frac{1}{2}} d u$

Step 10

Simplify.

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Step 10.1

Dividing two negative values results in a positive value.

$\frac{1}{3} (\ln (| y |) + C) - 5 (2 y^{\frac{1}{2}} + C) + \int - e^{u} \frac{1}{\frac{1}{2}} d u$

Step 10.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\frac{1}{3} (\ln (| y |) + C) - 5 (2 y^{\frac{1}{2}} + C) + \int - e^{u} (1 \cdot 2) d u$

Step 10.3

Multiply $2$ by $1$ .

$\frac{1}{3} (\ln (| y |) + C) - 5 (2 y^{\frac{1}{2}} + C) + \int - e^{u} \cdot 2 d u$

Step 10.4

Multiply $2$ by $- 1$ .

$\frac{1}{3} (\ln (| y |) + C) - 5 (2 y^{\frac{1}{2}} + C) + \int - 2 e^{u} d u$

Step 11

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

$\frac{1}{3} (\ln (| y |) + C) - 5 (2 y^{\frac{1}{2}} + C) - 2 \int e^{u} d u$

Step 12

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

$\frac{1}{3} (\ln (| y |) + C) - 5 (2 y^{\frac{1}{2}} + C) - 2 (e^{u} + C)$

Step 13

Simplify.

$\frac{\ln (| y |)}{3} - 10 y^{\frac{1}{2}} - 2 e^{u} + C$

Step 14

Replace all occurrences of $u$ with $- \frac{y}{2}$ .

$\frac{\ln (| y |)}{3} - 10 y^{\frac{1}{2}} - 2 e^{- \frac{y}{2}} + C$

Step 15

Reorder terms.

$\frac{1}{3} \ln (| y |) - 10 y^{\frac{1}{2}} - 2 e^{- \frac{1}{2} y} + C$