Calculus Examples

$\frac{d y}{d x} = - \frac{x^{2} + 1}{y \sqrt{y + 1}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y \sqrt{y + 1}$ .

$y \sqrt{y + 1} \frac{d y}{d x} = y \sqrt{y + 1} (- \frac{x^{2} + 1}{y \sqrt{y + 1}})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$y \sqrt{y + 1} \frac{d y}{d x} = - y \sqrt{y + 1} \frac{x^{2} + 1}{y \sqrt{y + 1}}$

Step 1.2.2

Cancel the common factor of $y \sqrt{y + 1}$ .

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Step 1.2.2.1

Factor $y \sqrt{y + 1}$ out of $- y \sqrt{y + 1}$ .

$y \sqrt{y + 1} \frac{d y}{d x} = y \sqrt{y + 1} (- 1) \frac{x^{2} + 1}{y \sqrt{y + 1}}$

Step 1.2.2.2

Cancel the common factor.

$y \sqrt{y + 1} \frac{d y}{d x} = y \sqrt{y + 1} \cdot - 1 \frac{x^{2} + 1}{y \sqrt{y + 1}}$

Step 1.2.2.3

Rewrite the expression.

$y \sqrt{y + 1} \frac{d y}{d x} = - (x^{2} + 1)$

Step 1.2.3

Apply the distributive property.

$y \sqrt{y + 1} \frac{d y}{d x} = - x^{2} - 1 \cdot 1$

Step 1.2.4

Multiply $- 1$ by $1$ .

$y \sqrt{y + 1} \frac{d y}{d x} = - x^{2} - 1$

Step 1.3

Rewrite the equation.

$y \sqrt{y + 1} d y = (- x^{2} - 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y \sqrt{y + 1} d y = \int - x^{2} - 1 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = y$ and $d v = \sqrt{y + 1}$ .

$y (\frac{2}{3} {(y + 1)}^{\frac{3}{2}}) - \int \frac{2}{3} {(y + 1)}^{\frac{3}{2}} d y = \int - x^{2} - 1 d x$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Combine $\frac{2}{3}$ and ${(y + 1)}^{\frac{3}{2}}$ .

$y \frac{2 {(y + 1)}^{\frac{3}{2}}}{3} - \int \frac{2}{3} {(y + 1)}^{\frac{3}{2}} d y = \int - x^{2} - 1 d x$

Step 2.2.2.2

Combine $y$ and $\frac{2 {(y + 1)}^{\frac{3}{2}}}{3}$ .

$\frac{y (2 {(y + 1)}^{\frac{3}{2}})}{3} - \int \frac{2}{3} {(y + 1)}^{\frac{3}{2}} d y = \int - x^{2} - 1 d x$

Step 2.2.3

Since $\frac{2}{3}$ is constant with respect to $y$ , move $\frac{2}{3}$ out of the integral.

$\frac{y (2 {(y + 1)}^{\frac{3}{2}})}{3} - (\frac{2}{3} \int {(y + 1)}^{\frac{3}{2}} d y) = \int - x^{2} - 1 d x$

Step 2.2.4

Let $u = y + 1$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.4.1

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

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Step 2.2.4.1.1

Differentiate $y + 1$ .

$\frac{d}{d y} [y + 1]$

Step 2.2.4.1.2

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

$\frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 2.2.4.1.3

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

$1 + \frac{d}{d y} [1]$

Step 2.2.4.1.4

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

$1 + 0$

Step 2.2.4.1.5

Add $1$ and $0$ .

$1$

Step 2.2.4.2

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

$\frac{y (2 {(y + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} \int u^{\frac{3}{2}} d u = \int - x^{2} - 1 d x$

Step 2.2.5

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

$\frac{y (2 {(y + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} (\frac{2}{5} u^{\frac{5}{2}} + C_{1}) = \int - x^{2} - 1 d x$

Step 2.2.6

Simplify.

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Step 2.2.6.1

Rewrite $\frac{y (2 {(y + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} (\frac{2}{5} u^{\frac{5}{2}} + C_{1})$ as $\frac{2}{3} y {(y + 1)}^{\frac{3}{2}} - \frac{2}{3} \cdot \frac{2}{5} u^{\frac{5}{2}} + C_{1}$ .

$\frac{2}{3} y {(y + 1)}^{\frac{3}{2}} - \frac{2}{3} \cdot \frac{2}{5} u^{\frac{5}{2}} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2

Simplify.

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Step 2.2.6.2.1

Combine $\frac{2}{3}$ and $y$ .

$\frac{2 y}{3} {(y + 1)}^{\frac{3}{2}} - \frac{2}{3} \cdot \frac{2}{5} u^{\frac{5}{2}} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.2

Combine $\frac{2 y}{3}$ and ${(y + 1)}^{\frac{3}{2}}$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot \frac{2}{5} u^{\frac{5}{2}} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.3

Multiply $\frac{2}{5}$ by $\frac{2}{3}$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}}}{3} - \frac{2 \cdot 2}{5 \cdot 3} u^{\frac{5}{2}} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.4

Multiply $2$ by $2$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}}}{3} - \frac{4}{5 \cdot 3} u^{\frac{5}{2}} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.5

Multiply $5$ by $3$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}}}{3} - \frac{4}{15} u^{\frac{5}{2}} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.6

To write $- \frac{4}{15} u^{\frac{5}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}}}{3} - \frac{4}{15} u^{\frac{5}{2}} \cdot \frac{3}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.7

Combine $- \frac{4}{15} u^{\frac{5}{2}}$ and $\frac{3}{3}$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}}}{3} + \frac{- \frac{4}{15} u^{\frac{5}{2}} \cdot 3}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.8

Combine the numerators over the common denominator.

$\frac{2 y {(y + 1)}^{\frac{3}{2}} - \frac{4}{15} u^{\frac{5}{2}} \cdot 3}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.9

Combine $u^{\frac{5}{2}}$ and $\frac{4}{15}$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}} - \frac{u^{\frac{5}{2}} \cdot 4}{15} \cdot 3}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.10

Multiply $3$ by $- 1$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}} - 3 \frac{u^{\frac{5}{2}} \cdot 4}{15}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.11

Combine $- 3$ and $\frac{u^{\frac{5}{2}} \cdot 4}{15}$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}} + \frac{- 3 (u^{\frac{5}{2}} \cdot 4)}{15}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.12

Multiply $4$ by $- 3$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}} + \frac{- 12 u^{\frac{5}{2}}}{15}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.13

Factor $3$ out of $- 12 u^{\frac{5}{2}}$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}} + \frac{3 (- 4 u^{\frac{5}{2}})}{15}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.14

Cancel the common factors.

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Step 2.2.6.2.14.1

Factor $3$ out of $15$ .

$\frac{2 y {(y + 1)}^{\frac{3}{2}} + \frac{3 (- 4 u^{\frac{5}{2}})}{3 (5)}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.14.2

Cancel the common factor.

$\frac{2 y {(y + 1)}^{\frac{3}{2}} + \frac{3 (- 4 u^{\frac{5}{2}})}{3 \cdot 5}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.14.3

Rewrite the expression.

$\frac{2 y {(y + 1)}^{\frac{3}{2}} + \frac{- 4 u^{\frac{5}{2}}}{5}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.15

Move the negative in front of the fraction.

$\frac{2 y {(y + 1)}^{\frac{3}{2}} - \frac{4 u^{\frac{5}{2}}}{5}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.16

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

$\frac{2 y {(y + 1)}^{\frac{3}{2}} \cdot \frac{5}{5} - \frac{4 u^{\frac{5}{2}}}{5}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.17

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

$\frac{\frac{2 y {(y + 1)}^{\frac{3}{2}} \cdot 5}{5} - \frac{4 u^{\frac{5}{2}}}{5}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.18

Combine the numerators over the common denominator.

$\frac{\frac{2 y {(y + 1)}^{\frac{3}{2}} \cdot 5 - 4 u^{\frac{5}{2}}}{5}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.19

Multiply $5$ by $2$ .

$\frac{\frac{10 y {(y + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5}}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.20

Rewrite $\frac{\frac{10 y {(y + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5}}{3}$ as a product.

$\frac{10 y {(y + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5} \cdot \frac{1}{3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.21

Multiply $\frac{10 y {(y + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5}$ by $\frac{1}{3}$ .

$\frac{10 y {(y + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5 \cdot 3} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.6.2.22

Multiply $5$ by $3$ .

$\frac{10 y {(y + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{15} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.7

Replace all occurrences of $u$ with $y + 1$ .

$\frac{10 y {(y + 1)}^{\frac{3}{2}} - 4 {(y + 1)}^{\frac{5}{2}}}{15} + C_{1} = \int - x^{2} - 1 d x$

Step 2.2.8

Reorder terms.

$\frac{1}{15} (10 y {(y + 1)}^{\frac{3}{2}} - 4 {(y + 1)}^{\frac{5}{2}}) + C_{1} = \int - x^{2} - 1 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$\frac{1}{15} (10 y {(y + 1)}^{\frac{3}{2}} - 4 {(y + 1)}^{\frac{5}{2}}) + C_{1} = \int - x^{2} d x + \int - 1 d x$

Step 2.3.2

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

$\frac{1}{15} (10 y {(y + 1)}^{\frac{3}{2}} - 4 {(y + 1)}^{\frac{5}{2}}) + C_{1} = - \int x^{2} d x + \int - 1 d x$

Step 2.3.3

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

$\frac{1}{15} (10 y {(y + 1)}^{\frac{3}{2}} - 4 {(y + 1)}^{\frac{5}{2}}) + C_{1} = - (\frac{1}{3} x^{3} + C_{2}) + \int - 1 d x$

Step 2.3.4

Apply the constant rule.

$\frac{1}{15} (10 y {(y + 1)}^{\frac{3}{2}} - 4 {(y + 1)}^{\frac{5}{2}}) + C_{1} = - (\frac{1}{3} x^{3} + C_{2}) - x + C_{3}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$\frac{1}{15} (10 y {(y + 1)}^{\frac{3}{2}} - 4 {(y + 1)}^{\frac{5}{2}}) + C_{1} = - (\frac{x^{3}}{3} + C_{2}) - x + C_{3}$

Step 2.3.5.2

Simplify.

$\frac{1}{15} (10 y {(y + 1)}^{\frac{3}{2}} - 4 {(y + 1)}^{\frac{5}{2}}) + C_{1} = - \frac{1}{3} x^{3} - x + C_{4}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{15} (10 y {(y + 1)}^{\frac{3}{2}} - 4 {(y + 1)}^{\frac{5}{2}}) = - \frac{1}{3} x^{3} - x + K$