Calculus Examples

$\int x^{2} \sqrt{2 + x} d x$

Step 1

Let $u_{1} = 2 + x$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 2 + x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $2 + x$ .

$\frac{d}{d x} [2 + x]$

Step 1.1.2

By the Sum Rule, the derivative of $2 + x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [x]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [x]$

Step 1.1.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x]$

Step 1.1.4

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

$0 + 1$

Step 1.1.5

Add $0$ and $1$ .

$1$

$1$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int {(u_{1} - 2)}^{2} \sqrt{u_{1}} d u_{1}$

$\int {(u_{1} - 2)}^{2} \sqrt{u_{1}} d u_{1}$

Step 2

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

$\int {(u_{1} - 2)}^{2} {u_{1}}^{\frac{1}{2}} d u_{1}$

Step 3

Let $u_{2} = u_{1} - 2$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 3.1

Let $u_{2} = u_{1} - 2$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 3.1.1

Differentiate $u_{1} - 2$ .

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

Step 3.1.2

By the Sum Rule, the derivative of $u_{1} - 2$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [- 2]$ .

$\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [- 2]$

Step 3.1.3

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

$1 + \frac{d}{d u_{1}} [- 2]$

Step 3.1.4

Since $- 2$ is constant with respect to $u_{1}$ , the derivative of $- 2$ with respect to $u_{1}$ is $0$ .

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

$1$

Step 3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int {u_{2}}^{2} {(u_{2} + 2)}^{\frac{1}{2}} d u_{2}$

$\int {u_{2}}^{2} {(u_{2} + 2)}^{\frac{1}{2}} d u_{2}$

Step 4

Let $u_{3} = u_{2} + 2$ . Then $d u_{3} = d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 4.1

Let $u_{3} = u_{2} + 2$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 4.1.1

Differentiate $u_{2} + 2$ .

$\frac{d}{d u_{2}} [u_{2} + 2]$

Step 4.1.2

By the Sum Rule, the derivative of $u_{2} + 2$ with respect to $u_{2}$ is $\frac{d}{d u_{2}} [u_{2}] + \frac{d}{d u_{2}} [2]$ .

$\frac{d}{d u_{2}} [u_{2}] + \frac{d}{d u_{2}} [2]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d u_{2}} [2]$

Step 4.1.4

Since $2$ is constant with respect to $u_{2}$ , the derivative of $2$ with respect to $u_{2}$ is $0$ .

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

$1$

Step 4.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\int {(u_{3} - 2)}^{2} {u_{3}}^{\frac{1}{2}} d u_{3}$

$\int {(u_{3} - 2)}^{2} {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5

Simplify.

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Step 5.1

Rewrite ${(u_{3} - 2)}^{2}$ as $(u_{3} - 2) (u_{3} - 2)$ .

$\int (u_{3} - 2) (u_{3} - 2) {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.2

Apply the distributive property.

$\int (u_{3} (u_{3} - 2) - 2 (u_{3} - 2)) {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.3

Apply the distributive property.

$\int (u_{3} \cdot u_{3} + u_{3} \cdot - 2 - 2 (u_{3} - 2)) {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.4

Apply the distributive property.

$\int (u_{3} \cdot u_{3} + u_{3} \cdot - 2 - 2 u_{3} - 2 \cdot - 2) {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.5

Apply the distributive property.

$\int (u_{3} \cdot u_{3} + u_{3} \cdot - 2) {u_{3}}^{\frac{1}{2}} + (- 2 u_{3} - 2 \cdot - 2) {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.6

Apply the distributive property.

$\int u_{3} \cdot u_{3} \cdot {u_{3}}^{\frac{1}{2}} + u_{3} \cdot - 2 {u_{3}}^{\frac{1}{2}} + (- 2 u_{3} - 2 \cdot - 2) {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.7

Apply the distributive property.

$\int u_{3} \cdot u_{3} \cdot {u_{3}}^{\frac{1}{2}} + u_{3} \cdot - 2 {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.8

Reorder $u_{3}$ and $- 2$ .

$\int u_{3} \cdot u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.9

Raise $u_{3}$ to the power of $1$ .

$\int {u_{3}}^{1} u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.10

Raise $u_{3}$ to the power of $1$ .

$\int {u_{3}}^{1} {u_{3}}^{1} {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.11

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

$\int {u_{3}}^{1 + 1} {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.12

Add $1$ and $1$ .

$\int {u_{3}}^{2} {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.13

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

$\int {u_{3}}^{2 + \frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.14

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int {u_{3}}^{2 \cdot \frac{2}{2} + \frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.15

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

$\int {u_{3}}^{\frac{2 \cdot 2}{2} + \frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.16

Combine the numerators over the common denominator.

$\int {u_{3}}^{\frac{2 \cdot 2 + 1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.17

Simplify the numerator.

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Step 5.17.1

Multiply $2$ by $2$ .

$\int {u_{3}}^{\frac{4 + 1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.17.2

Add $4$ and $1$ .

$\int {u_{3}}^{\frac{5}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

$\int {u_{3}}^{\frac{5}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.18

Raise $u_{3}$ to the power of $1$ .

$\int {u_{3}}^{\frac{5}{2}} - 2 ({u_{3}}^{1} {u_{3}}^{\frac{1}{2}}) - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.19

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

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{1 + \frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.20

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

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{\frac{2}{2} + \frac{1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.21

Combine the numerators over the common denominator.

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{\frac{2 + 1}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.22

Add $2$ and $1$ .

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{\frac{3}{2}} - 2 u_{3} \cdot {u_{3}}^{\frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.23

Raise $u_{3}$ to the power of $1$ .

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{\frac{3}{2}} - 2 ({u_{3}}^{1} {u_{3}}^{\frac{1}{2}}) - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.24

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

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{\frac{3}{2}} - 2 {u_{3}}^{1 + \frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.25

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

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{\frac{3}{2}} - 2 {u_{3}}^{\frac{2}{2} + \frac{1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.26

Combine the numerators over the common denominator.

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{\frac{3}{2}} - 2 {u_{3}}^{\frac{2 + 1}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.27

Add $2$ and $1$ .

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{\frac{3}{2}} - 2 {u_{3}}^{\frac{3}{2}} - 2 \cdot - 2 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.28

Multiply $- 2$ by $- 2$ .

$\int {u_{3}}^{\frac{5}{2}} - 2 {u_{3}}^{\frac{3}{2}} - 2 {u_{3}}^{\frac{3}{2}} + 4 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.29

Subtract $2 {u_{3}}^{\frac{3}{2}}$ from $- 2 {u_{3}}^{\frac{3}{2}}$ .

$\int {u_{3}}^{\frac{5}{2}} - 4 {u_{3}}^{\frac{3}{2}} + 4 {u_{3}}^{\frac{1}{2}} d u_{3}$

Step 5.30

Reorder $- 4 {u_{3}}^{\frac{3}{2}}$ and $4 {u_{3}}^{\frac{1}{2}}$ .

$\int {u_{3}}^{\frac{5}{2}} + 4 {u_{3}}^{\frac{1}{2}} - 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 5.31

Reorder ${u_{3}}^{\frac{5}{2}}$ and $4 {u_{3}}^{\frac{1}{2}}$ .

$\int 4 {u_{3}}^{\frac{1}{2}} + {u_{3}}^{\frac{5}{2}} - 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

$\int 4 {u_{3}}^{\frac{1}{2}} + {u_{3}}^{\frac{5}{2}} - 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 6

Split the single integral into multiple integrals.

$\int 4 {u_{3}}^{\frac{1}{2}} d u_{3} + \int {u_{3}}^{\frac{5}{2}} d u_{3} + \int - 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 7

Since $4$ is constant with respect to $u_{3}$ , move $4$ out of the integral.

$4 \int {u_{3}}^{\frac{1}{2}} d u_{3} + \int {u_{3}}^{\frac{5}{2}} d u_{3} + \int - 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 8

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

$4 (\frac{2}{3} {u_{3}}^{\frac{3}{2}} + C) + \int {u_{3}}^{\frac{5}{2}} d u_{3} + \int - 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 9

By the Power Rule, the integral of ${u_{3}}^{\frac{5}{2}}$ with respect to $u_{3}$ is $\frac{2}{7} {u_{3}}^{\frac{7}{2}}$ .

$4 (\frac{2}{3} {u_{3}}^{\frac{3}{2}} + C) + \frac{2}{7} {u_{3}}^{\frac{7}{2}} + C + \int - 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 10

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

$4 (\frac{2 {u_{3}}^{\frac{3}{2}}}{3} + C) + \frac{2}{7} {u_{3}}^{\frac{7}{2}} + C + \int - 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 11

Since $- 4$ is constant with respect to $u_{3}$ , move $- 4$ out of the integral.

$4 (\frac{2 {u_{3}}^{\frac{3}{2}}}{3} + C) + \frac{2}{7} {u_{3}}^{\frac{7}{2}} + C - 4 \int {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 12

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

$4 (\frac{2 {u_{3}}^{\frac{3}{2}}}{3} + C) + \frac{2}{7} {u_{3}}^{\frac{7}{2}} + C - 4 (\frac{2}{5} {u_{3}}^{\frac{5}{2}} + C)$

Step 13

Simplify.

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Step 13.1

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

$4 (\frac{2 {u_{3}}^{\frac{3}{2}}}{3} + C) + \frac{2}{7} {u_{3}}^{\frac{7}{2}} + C - 4 (\frac{2 {u_{3}}^{\frac{5}{2}}}{5} + C)$

Step 13.2

Simplify.

$\frac{8 {u_{3}}^{\frac{3}{2}}}{3} + \frac{2 {u_{3}}^{\frac{7}{2}}}{7} - 4 \frac{2 {u_{3}}^{\frac{5}{2}}}{5} + C$

$\frac{8 {u_{3}}^{\frac{3}{2}}}{3} + \frac{2 {u_{3}}^{\frac{7}{2}}}{7} - 4 \frac{2 {u_{3}}^{\frac{5}{2}}}{5} + C$

Step 14

Reorder terms.

$\frac{8}{3} {u_{3}}^{\frac{3}{2}} + \frac{2}{7} {u_{3}}^{\frac{7}{2}} - 4 (\frac{2}{5}) {u_{3}}^{\frac{5}{2}} + C$

Step 15

Simplify.

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Step 15.1

Combine $- 4$ and $\frac{2}{5}$ .

$\frac{8}{3} {u_{3}}^{\frac{3}{2}} + \frac{2}{7} {u_{3}}^{\frac{7}{2}} + \frac{- 4 \cdot 2}{5} {u_{3}}^{\frac{5}{2}} + C$

Step 15.2

Multiply $- 4$ by $2$ .

$\frac{8}{3} {u_{3}}^{\frac{3}{2}} + \frac{2}{7} {u_{3}}^{\frac{7}{2}} + \frac{- 8}{5} {u_{3}}^{\frac{5}{2}} + C$

Step 15.3

Move the negative in front of the fraction.

$\frac{8}{3} {u_{3}}^{\frac{3}{2}} + \frac{2}{7} {u_{3}}^{\frac{7}{2}} - \frac{8}{5} {u_{3}}^{\frac{5}{2}} + C$

$\frac{8}{3} {u_{3}}^{\frac{3}{2}} + \frac{2}{7} {u_{3}}^{\frac{7}{2}} - \frac{8}{5} {u_{3}}^{\frac{5}{2}} + C$

Step 16

Substitute back in for each integration substitution variable.

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Step 16.1

Replace all occurrences of $u_{3}$ with $u_{2} + 2$ .

$\frac{8}{3} {(u_{2} + 2)}^{\frac{3}{2}} + \frac{2}{7} {(u_{2} + 2)}^{\frac{7}{2}} - \frac{8}{5} {(u_{2} + 2)}^{\frac{5}{2}} + C$

Step 16.2

Replace all occurrences of $u_{2}$ with $u_{1} - 2$ .

$\frac{8}{3} {(u_{1} - 2 + 2)}^{\frac{3}{2}} + \frac{2}{7} {(u_{1} - 2 + 2)}^{\frac{7}{2}} - \frac{8}{5} {(u_{1} - 2 + 2)}^{\frac{5}{2}} + C$

Step 16.3

Replace all occurrences of $u_{1}$ with $2 + x$ .

$\frac{8}{3} {(2 + x - 2 + 2)}^{\frac{3}{2}} + \frac{2}{7} {(2 + x - 2 + 2)}^{\frac{7}{2}} - \frac{8}{5} {(2 + x - 2 + 2)}^{\frac{5}{2}} + C$

$\frac{8}{3} {(2 + x - 2 + 2)}^{\frac{3}{2}} + \frac{2}{7} {(2 + x - 2 + 2)}^{\frac{7}{2}} - \frac{8}{5} {(2 + x - 2 + 2)}^{\frac{5}{2}} + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$