Calculus Examples

$\int x^{3} \sqrt{x - 4} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{3}$ and $d v = \sqrt{x - 4}$ .

$x^{3} (\frac{2}{3} {(x - 4)}^{\frac{3}{2}}) - \int \frac{2}{3} {(x - 4)}^{\frac{3}{2}} (3 x^{2}) d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{2}{3}$ and ${(x - 4)}^{\frac{3}{2}}$ .

$x^{3} \frac{2 {(x - 4)}^{\frac{3}{2}}}{3} - \int \frac{2}{3} {(x - 4)}^{\frac{3}{2}} (3 x^{2}) d x$

Step 2.2

Combine $x^{3}$ and $\frac{2 {(x - 4)}^{\frac{3}{2}}}{3}$ .

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - \int \frac{2}{3} {(x - 4)}^{\frac{3}{2}} (3 x^{2}) d x$

Step 3

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

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - (\frac{2}{3} \cdot 3 \int {(x - 4)}^{\frac{3}{2}} (x^{2}) d x)$

Step 4

Simplify.

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Step 4.1

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

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - (\frac{2 \cdot 3}{3} \int {(x - 4)}^{\frac{3}{2}} (x^{2}) d x)$

Step 4.2

Multiply $2$ by $3$ .

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - (\frac{6}{3} \int {(x - 4)}^{\frac{3}{2}} (x^{2}) d x)$

Step 4.3

Cancel the common factor of $6$ and $3$ .

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Step 4.3.1

Factor $3$ out of $6$ .

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - (\frac{3 \cdot 2}{3} \int {(x - 4)}^{\frac{3}{2}} (x^{2}) d x)$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $3$ out of $3$ .

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - (\frac{3 \cdot 2}{3 (1)} \int {(x - 4)}^{\frac{3}{2}} (x^{2}) d x)$

Step 4.3.2.2

Cancel the common factor.

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - (\frac{3 \cdot 2}{3 \cdot 1} \int {(x - 4)}^{\frac{3}{2}} (x^{2}) d x)$

Step 4.3.2.3

Rewrite the expression.

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

Step 4.3.2.4

Divide $2$ by $1$ .

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - (2 \int {(x - 4)}^{\frac{3}{2}} (x^{2}) d x)$

Step 4.4

Multiply $2$ by $- 1$ .

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - 2 \int {(x - 4)}^{\frac{3}{2}} (x^{2}) d x$

Step 5

Let $u_{1} = x - 4$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 5.1

Let $u_{1} = x - 4$ . Find $\frac{d u_{1}}{d x}$ .

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Step 5.1.1

Differentiate $x - 4$ .

$\frac{d}{d x} [x - 4]$

Step 5.1.2

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$

Step 5.1.3

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

$1 + \frac{d}{d x} [- 4]$

Step 5.1.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

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

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

Step 6

Let $u_{2} = u_{1} + 4$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.1

Let $u_{2} = u_{1} + 4$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 6.1.1

Differentiate $u_{1} + 4$ .

$\frac{d}{d u_{1}} [u_{1} + 4]$

Step 6.1.2

By the Sum Rule, the derivative of $u_{1} + 4$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [4]$ .

$\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [4]$

Step 6.1.3

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

$1 + \frac{d}{d u_{1}} [4]$

Step 6.1.4

Since $4$ is constant with respect to $u_{1}$ , the derivative of $4$ with respect to $u_{1}$ is $0$ .

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

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

$\frac{x^{3} (2 {(x - 4)}^{\frac{3}{2}})}{3} - 2 \int {(u_{2} - 4)}^{\frac{3}{2}} {u_{2}}^{2} d u_{2}$

Step 7

Let $u_{3} = u_{2} - 4$ . Then $d u_{3} = d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 7.1

Let $u_{3} = u_{2} - 4$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 7.1.1

Differentiate $u_{2} - 4$ .

$\frac{d}{d u_{2}} [u_{2} - 4]$

Step 7.1.2

By the Sum Rule, the derivative of $u_{2} - 4$ with respect to $u_{2}$ is $\frac{d}{d u_{2}} [u_{2}] + \frac{d}{d u_{2}} [- 4]$ .

$\frac{d}{d u_{2}} [u_{2}] + \frac{d}{d u_{2}} [- 4]$

Step 7.1.3

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

$1 + \frac{d}{d u_{2}} [- 4]$

Step 7.1.4

Since $- 4$ is constant with respect to $u_{2}$ , the derivative of $- 4$ with respect to $u_{2}$ is $0$ .

$1 + 0$

Step 7.1.5

Add $1$ and $0$ .

$1$

Step 7.2

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

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

Step 8

Simplify.

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Step 8.1

Rewrite ${(u_{3} + 4)}^{2}$ as $(u_{3} + 4) (u_{3} + 4)$ .

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

Step 8.2