Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x - 2$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

$1$

Step 1.2

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

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

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

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $u_{1} + 2$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

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

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

$1$

Step 2.2

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

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

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

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $u_{2} - 2$ .

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

Step 3.1.2

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 3.1.3

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 3.1.4

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

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

$1$

Step 3.2

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

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Apply the distributive property.

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

Step 4.6

Apply the distributive property.

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

Step 4.7

Apply the distributive property.

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

Add $1$ and $1$ .

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

Step 4.13

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

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

Step 4.14

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

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

Step 4.15

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

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

Step 4.16

Combine the numerators over the common denominator.

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

Step 4.17

Simplify the numerator.

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Step 4.17.1

Multiply $2$ by $2$ .

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

Step 4.17.2

Add $4$ and $3$ .

$\int {u_{3}}^{\frac{7}{2}} + 2 u_{3} \cdot {u_{3}}^{\frac{3}{2}} + 2 u_{3} \cdot {u_{3}}^{\frac{3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

$\int {u_{3}}^{\frac{7}{2}} + 2 u_{3} \cdot {u_{3}}^{\frac{3}{2}} + 2 u_{3} \cdot {u_{3}}^{\frac{3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.18

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

$\int {u_{3}}^{\frac{7}{2}} + 2 ({u_{3}}^{1} {u_{3}}^{\frac{3}{2}}) + 2 u_{3} \cdot {u_{3}}^{\frac{3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.19

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

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{1 + \frac{3}{2}} + 2 u_{3} \cdot {u_{3}}^{\frac{3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.20

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

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{\frac{2}{2} + \frac{3}{2}} + 2 u_{3} \cdot {u_{3}}^{\frac{3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.21

Combine the numerators over the common denominator.

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{\frac{2 + 3}{2}} + 2 u_{3} \cdot {u_{3}}^{\frac{3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.22

Add $2$ and $3$ .

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{\frac{5}{2}} + 2 u_{3} \cdot {u_{3}}^{\frac{3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.23

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

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{\frac{5}{2}} + 2 ({u_{3}}^{1} {u_{3}}^{\frac{3}{2}}) + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.24

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

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{\frac{5}{2}} + 2 {u_{3}}^{1 + \frac{3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.25

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

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{\frac{5}{2}} + 2 {u_{3}}^{\frac{2}{2} + \frac{3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.26

Combine the numerators over the common denominator.

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{\frac{5}{2}} + 2 {u_{3}}^{\frac{2 + 3}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.27

Add $2$ and $3$ .

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{\frac{5}{2}} + 2 {u_{3}}^{\frac{5}{2}} + 2 \cdot 2 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.28

Multiply $2$ by $2$ .

$\int {u_{3}}^{\frac{7}{2}} + 2 {u_{3}}^{\frac{5}{2}} + 2 {u_{3}}^{\frac{5}{2}} + 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.29

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

$\int {u_{3}}^{\frac{7}{2}} + 4 {u_{3}}^{\frac{5}{2}} + 4 {u_{3}}^{\frac{3}{2}} d u_{3}$

Step 4.30

Move ${u_{3}}^{\frac{7}{2}}$ .

$\int 4 {u_{3}}^{\frac{5}{2}} + 4 {u_{3}}^{\frac{3}{2}} + {u_{3}}^{\frac{7}{2}} d u_{3}$

$\int 4 {u_{3}}^{\frac{5}{2}} + 4 {u_{3}}^{\frac{3}{2}} + {u_{3}}^{\frac{7}{2}} d u_{3}$

Step 5

Split the single integral into multiple integrals.

$\int 4 {u_{3}}^{\frac{5}{2}} d u_{3} + \int 4 {u_{3}}^{\frac{3}{2}} d u_{3} + \int {u_{3}}^{\frac{7}{2}} d u_{3}$

Step 6

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

$4 \int {u_{3}}^{\frac{5}{2}} d u_{3} + \int 4 {u_{3}}^{\frac{3}{2}} d u_{3} + \int {u_{3}}^{\frac{7}{2}} d u_{3}$

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

By the Power Rule, the integral of ${u_{3}}^{\frac{7}{2}}$ with respect to $u_{3}$ is $\frac{2}{9} {u_{3}}^{\frac{9}{2}}$ .

$4 (\frac{2}{7} {u_{3}}^{\frac{7}{2}} + C) + 4 (\frac{2}{5} {u_{3}}^{\frac{5}{2}} + C) + \frac{2}{9} {u_{3}}^{\frac{9}{2}} + C$

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

Combine $\frac{2}{7}$ and ${u_{3}}^{\frac{7}{2}}$ .

$4 (\frac{2 {u_{3}}^{\frac{7}{2}}}{7} + C) + 4 (\frac{2}{5} {u_{3}}^{\frac{5}{2}} + C) + \frac{2}{9} {u_{3}}^{\frac{9}{2}} + C$

Step 11.1.2

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

$4 (\frac{2 {u_{3}}^{\frac{7}{2}}}{7} + C) + 4 (\frac{2 {u_{3}}^{\frac{5}{2}}}{5} + C) + \frac{2}{9} {u_{3}}^{\frac{9}{2}} + C$

$4 (\frac{2 {u_{3}}^{\frac{7}{2}}}{7} + C) + 4 (\frac{2 {u_{3}}^{\frac{5}{2}}}{5} + C) + \frac{2}{9} {u_{3}}^{\frac{9}{2}} + C$

Step 11.2

Simplify.

$\frac{8 {u_{3}}^{\frac{7}{2}}}{7} + \frac{8 {u_{3}}^{\frac{5}{2}}}{5} + \frac{2}{9} {u_{3}}^{\frac{9}{2}} + C$

$\frac{8 {u_{3}}^{\frac{7}{2}}}{7} + \frac{8 {u_{3}}^{\frac{5}{2}}}{5} + \frac{2}{9} {u_{3}}^{\frac{9}{2}} + C$

Step 12

Substitute back in for each integration substitution variable.

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Step 12.1

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

$\frac{8 {(u_{2} - 2)}^{\frac{7}{2}}}{7} + \frac{8 {(u_{2} - 2)}^{\frac{5}{2}}}{5} + \frac{2}{9} {(u_{2} - 2)}^{\frac{9}{2}} + C$

Step 12.2

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

$\frac{8 {(u_{1} + 2 - 2)}^{\frac{7}{2}}}{7} + \frac{8 {(u_{1} + 2 - 2)}^{\frac{5}{2}}}{5} + \frac{2}{9} {(u_{1} + 2 - 2)}^{\frac{9}{2}} + C$

Step 12.3

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

$\frac{8 {(x - 2 + 2 - 2)}^{\frac{7}{2}}}{7} + \frac{8 {(x - 2 + 2 - 2)}^{\frac{5}{2}}}{5} + \frac{2}{9} {(x - 2 + 2 - 2)}^{\frac{9}{2}} + C$

$\frac{8 {(x - 2 + 2 - 2)}^{\frac{7}{2}}}{7} + \frac{8 {(x - 2 + 2 - 2)}^{\frac{5}{2}}}{5} + \frac{2}{9} {(x - 2 + 2 - 2)}^{\frac{9}{2}} + C$

Step 13

Reorder terms.

$\frac{8}{7} {(x - 2 + 2 - 2)}^{\frac{7}{2}} + \frac{8}{5} {(x - 2 + 2 - 2)}^{\frac{5}{2}} + \frac{2}{9} {(x - 2 + 2 - 2)}^{\frac{9}{2}} + C$

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