Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x - 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

Differentiate $u_{1} + 1$ .

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

Step 3.1.2

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

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

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 $n = 1$ .

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

Step 3.1.4

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

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

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

Step 4

Let $u_{3} = u_{2} - 1$ . Then $d u_{3} = d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 4.1

Let $u_{3} = u_{2} - 1$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 4.1.1

Differentiate $u_{2} - 1$ .

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

Step 4.1.2

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

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

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 $n = 1$ .

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

Step 4.1.4

Since $- 1$ is constant with respect to $u_{2}$ , the derivative of $- 1$ 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 $d u_{3}$ .

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

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

Step 5

Simplify.

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Step 5.1

Rewrite ${(u_{3} + 1)}^{2}$ as $(u_{3} + 1) (u_{3} + 1)$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Apply the distributive property.

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

Step 5.6

Apply the distributive property.

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

Step 5.7

Apply the distributive property.

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

Step 5.8

Reorder $u_{3}$ and $1$ .

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

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

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

Step 5.12

Add $1$ and $1$ .

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

Step 5.13

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

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

Step 5.14

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

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

Step 5.15

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

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

Step 5.16

Combine the numerators over the common denominator.

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

Step 5.17

Simplify the numerator.

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Step 5.17.1

Multiply $2$ by $2$ .

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

Step 5.17.2

Add $4$ and $1$ .

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

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

Step 5.18

Multiply $u_{3}$ by $1$ .

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

Step 5.19

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

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

Step 5.20

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

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

Step 5.21

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

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

Step 5.22

Combine the numerators over the common denominator.

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

Step 5.23

Add $2$ and $1$ .

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

Step 5.24

Multiply $u_{3}$ by $1$ .

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

Step 5.25

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

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

Step 5.26

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

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

Step 5.27

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

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

Step 5.28

Combine the numerators over the common denominator.

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

Step 5.29

Add $2$ and $1$ .

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

Step 5.30

Multiply $1$ by $1$ .

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

Step 5.31

Multiply ${u_{3}}^{\frac{1}{2}}$ by $1$ .

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

Step 5.32

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

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

Step 5.33

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

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

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Simplify.

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

Step 12.2

Reorder terms.

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

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

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

Replace all occurrences of $u_{3}$ with $u_{2} - 1$ .

$\frac{4}{5} {(u_{2} - 1)}^{\frac{5}{2}} + \frac{2}{7} {(u_{2} - 1)}^{\frac{7}{2}} + \frac{2}{3} {(u_{2} - 1)}^{\frac{3}{2}} + C$

Step 13.2

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

$\frac{4}{5} {(u_{1} + 1 - 1)}^{\frac{5}{2}} + \frac{2}{7} {(u_{1} + 1 - 1)}^{\frac{7}{2}} + \frac{2}{3} {(u_{1} + 1 - 1)}^{\frac{3}{2}} + C$

Step 13.3

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

$\frac{4}{5} {(x - 1 + 1 - 1)}^{\frac{5}{2}} + \frac{2}{7} {(x - 1 + 1 - 1)}^{\frac{7}{2}} + \frac{2}{3} {(x - 1 + 1 - 1)}^{\frac{3}{2}} + C$

$\frac{4}{5} {(x - 1 + 1 - 1)}^{\frac{5}{2}} + \frac{2}{7} {(x - 1 + 1 - 1)}^{\frac{7}{2}} + \frac{2}{3} {(x - 1 + 1 - 1)}^{\frac{3}{2}} + C$

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