Calculus Examples

$\int \frac{y}{\sqrt{5 - y}} d y$

Step 1

Let $u = 5 - y$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $5 - y$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$\frac{d}{d y} [5] + \frac{d}{d y} [- y]$

Step 1.1.2.2

Since $5$ is constant with respect to $y$ , the derivative of $5$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [- y]$

Step 1.1.3

Evaluate $\frac{d}{d y} [- y]$ .

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Step 1.1.3.1

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

$0 - \frac{d}{d y} [y]$

Step 1.1.3.2

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

$0 - 1 \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 1.1.4

Subtract $1$ from $0$ .

$- 1$

Step 1.2

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

$\int \frac{- (- u + 5)}{\sqrt{u}} d u$

Step 2

Move the negative in front of the fraction.

$\int - \frac{- u + 5}{\sqrt{u}} d u$

Step 3

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

$- \int \frac{- u + 5}{\sqrt{u}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

Expand $(- u + 5) u^{- \frac{1}{2}}$ .

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Step 5.1

Apply the distributive property.

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

Step 5.2

Factor out negative.

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Write $1$ as a fraction with a common denominator.

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

Step 5.6

Combine the numerators over the common denominator.

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

Step 5.7

Subtract $1$ from $2$ .

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

Step 5.8

Reorder $- u^{\frac{1}{2}}$ and $5 u^{- \frac{1}{2}}$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

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

Step 11

Simplify.

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

Step 12

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

$- (10 {(5 - y)}^{\frac{1}{2}} - \frac{2 {(5 - y)}^{\frac{3}{2}}}{3}) + C$

Step 13

Simplify.

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Step 13.1

To write $10 {(5 - y)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- (10 {(5 - y)}^{\frac{1}{2}} \cdot \frac{3}{3} - \frac{2 {(5 - y)}^{\frac{3}{2}}}{3}) + C$

Step 13.2

Combine $10 {(5 - y)}^{\frac{1}{2}}$ and $\frac{3}{3}$ .

$- (\frac{10 {(5 - y)}^{\frac{1}{2}} \cdot 3}{3} - \frac{2 {(5 - y)}^{\frac{3}{2}}}{3}) + C$

Step 13.3

Combine the numerators over the common denominator.

$- \frac{10 {(5 - y)}^{\frac{1}{2}} \cdot 3 - 2 {(5 - y)}^{\frac{3}{2}}}{3} + C$

Step 13.4

Multiply $3$ by $10$ .

$- \frac{30 {(5 - y)}^{\frac{1}{2}} - 2 {(5 - y)}^{\frac{3}{2}}}{3} + C$

Step 14

Reorder terms.

$- \frac{1}{3} (30 {(5 - y)}^{\frac{1}{2}} - 2 {(5 - y)}^{\frac{3}{2}}) + C$