Calculus Examples

$\frac{d y}{d x} = x^{2} \sqrt{x - 3}$

Step 1

Rewrite the equation.

$d y = x^{2} \sqrt{x - 3} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x^{2} \sqrt{x - 3} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int x^{2} \sqrt{x - 3} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \sqrt{x - 3}$ .

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Combine $\frac{2}{3}$ and ${(x - 3)}^{\frac{3}{2}}$ .

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

Step 2.3.2.2

Combine $x^{2}$ and $\frac{2 {(x - 3)}^{\frac{3}{2}}}{3}$ .

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

Step 2.3.3

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

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

Step 2.3.4

Simplify.

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Step 2.3.4.1

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

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

Step 2.3.4.2

Multiply $2$ by $2$ .

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

Step 2.3.5

Let $u = x - 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.5.1

Let $u = x - 3$ . Find $\frac{d u}{d x}$ .

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Step 2.3.5.1.1

Differentiate $x - 3$ .

$\frac{d}{d x} [x - 3]$

Step 2.3.5.1.2

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 2.3.5.1.3

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

$1 + \frac{d}{d x} [- 3]$

Step 2.3.5.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.5.1.5

Add $1$ and $0$ .

$1$

Step 2.3.5.2

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

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

Step 2.3.6

Expand $u^{\frac{3}{2}} (u + 3)$ .

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Step 2.3.6.1

Apply the distributive property.

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

Step 2.3.6.2

Reorder $u^{\frac{3}{2}}$ and $3$ .

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

Step 2.3.6.3

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

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

Step 2.3.6.4

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

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

Step 2.3.6.5

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

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

Step 2.3.6.6

Combine the numerators over the common denominator.

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

Step 2.3.6.7

Add $3$ and $2$ .

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

Step 2.3.6.8

Reorder $u^{\frac{5}{2}}$ and $3 u^{\frac{3}{2}}$ .

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

Step 2.3.7

Split the single integral into multiple integrals.

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

Step 2.3.8

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

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

Step 2.3.9

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

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

Step 2.3.10

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

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

Step 2.3.11

By the Power Rule, the integral of $u^{\frac{5}{2}}$ with respect to $u$ is $\frac{2}{7} u^{\frac{7}{2}}$ .

$y + C_{1} = \frac{x^{2} (2 {(x - 3)}^{\frac{3}{2}})}{3} - \frac{4}{3} (3 (\frac{2 u^{\frac{5}{2}}}{5} + C_{2}) + \frac{2}{7} u^{\frac{7}{2}} + C_{3})$

Step 2.3.12

Simplify.

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Step 2.3.12.1

Combine $\frac{2}{7}$ and $u^{\frac{7}{2}}$ .

$y + C_{1} = \frac{x^{2} (2 {(x - 3)}^{\frac{3}{2}})}{3} - \frac{4}{3} (3 (\frac{2 u^{\frac{5}{2}}}{5} + C_{2}) + \frac{2 u^{\frac{7}{2}}}{7} + C_{3})$

Step 2.3.12.2

Simplify.

$y + C_{1} = \frac{2}{3} x^{2} {(x - 3)}^{\frac{3}{2}} - \frac{4}{3} (3 (\frac{2}{5}) u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}}) + C_{4}$

Step 2.3.12.3

Simplify.

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Step 2.3.12.3.1

Combine $\frac{2}{3}$ and $x^{2}$ .

$y + C_{1} = \frac{2 x^{2}}{3} {(x - 3)}^{\frac{3}{2}} - \frac{4}{3} (3 (\frac{2}{5}) u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}}) + C_{4}$

Step 2.3.12.3.2

Combine $\frac{2 x^{2}}{3}$ and ${(x - 3)}^{\frac{3}{2}}$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}}}{3} - \frac{4}{3} (3 (\frac{2}{5}) u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}}) + C_{4}$

Step 2.3.12.3.3

Combine $3$ and $\frac{2}{5}$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}}}{3} - \frac{4}{3} (\frac{3 \cdot 2}{5} u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}}) + C_{4}$

Step 2.3.12.3.4

Multiply $3$ by $2$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}}}{3} - \frac{4}{3} (\frac{6}{5} u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}}) + C_{4}$

Step 2.3.12.3.5

To write $- \frac{4}{3} (\frac{6}{5} u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}}}{3} - \frac{4}{3} (\frac{6}{5} u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}}) \cdot \frac{3}{3} + C_{4}$

Step 2.3.12.3.6

Combine $- \frac{4}{3} (\frac{6}{5} u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}})$ and $\frac{3}{3}$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}}}{3} + \frac{- \frac{4}{3} (\frac{6}{5} u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}}) \cdot 3}{3} + C_{4}$

Step 2.3.12.3.7

Combine the numerators over the common denominator.

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} - \frac{4}{3} (\frac{6}{5} u^{\frac{5}{2}} + \frac{2}{7} u^{\frac{7}{2}}) \cdot 3}{3} + C_{4}$

Step 2.3.12.3.8

Combine $\frac{6}{5}$ and $u^{\frac{5}{2}}$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} - \frac{4}{3} (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2}{7} u^{\frac{7}{2}}) \cdot 3}{3} + C_{4}$

Step 2.3.12.3.9

Combine $\frac{2}{7}$ and $u^{\frac{7}{2}}$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} - \frac{4}{3} (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{7}{2}}}{7}) \cdot 3}{3} + C_{4}$

Step 2.3.12.3.10

Multiply $3$ by $- 1$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} - 3 (\frac{4}{3}) (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{7}{2}}}{7})}{3} + C_{4}$

Step 2.3.12.3.11

Combine $- 3$ and $\frac{4}{3}$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} + \frac{- 3 \cdot 4}{3} (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{7}{2}}}{7})}{3} + C_{4}$

Step 2.3.12.3.12

Multiply $- 3$ by $4$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} + \frac{- 12}{3} (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{7}{2}}}{7})}{3} + C_{4}$

Step 2.3.12.3.13

Cancel the common factor of $- 12$ and $3$ .

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Step 2.3.12.3.13.1

Factor $3$ out of $- 12$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} + \frac{3 \cdot - 4}{3} (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{7}{2}}}{7})}{3} + C_{4}$

Step 2.3.12.3.13.2

Cancel the common factors.

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Step 2.3.12.3.13.2.1

Factor $3$ out of $3$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} + \frac{3 \cdot - 4}{3 (1)} (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{7}{2}}}{7})}{3} + C_{4}$

Step 2.3.12.3.13.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} + \frac{3 \cdot - 4}{3 \cdot 1} (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{7}{2}}}{7})}{3} + C_{4}$

Step 2.3.12.3.13.2.3

Rewrite the expression.

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} + \frac{- 4}{1} (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{7}{2}}}{7})}{3} + C_{4}$

Step 2.3.12.3.13.2.4

Divide $- 4$ by $1$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} - 4 (\frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{7}{2}}}{7})}{3} + C_{4}$

Step 2.3.13

Replace all occurrences of $u$ with $x - 3$ .

$y + C_{1} = \frac{2 x^{2} {(x - 3)}^{\frac{3}{2}} - 4 (\frac{6 {(x - 3)}^{\frac{5}{2}}}{5} + \frac{2 {(x - 3)}^{\frac{7}{2}}}{7})}{3} + C_{4}$

Step 2.3.14

Reorder terms.

$y + C_{1} = \frac{1}{3} (2 x^{2} {(x - 3)}^{\frac{3}{2}} - 4 (\frac{6}{5} {(x - 3)}^{\frac{5}{2}} + \frac{2}{7} {(x - 3)}^{\frac{7}{2}})) + C_{4}$

Step 2.4

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

$y = \frac{1}{3} (2 x^{2} {(x - 3)}^{\frac{3}{2}} - 4 (\frac{6}{5} {(x - 3)}^{\frac{5}{2}} + \frac{2}{7} {(x - 3)}^{\frac{7}{2}})) + K$