Calculus Examples

$\int x^{5} \sqrt{1 + x^{2}} d x$

Step 1

Let $u_{1} = 1 + x^{2}$ . Then $d u_{1} = 2 x d x$ , so $\frac{1}{2} d u_{1} = x d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $1 + x^{2}$ .

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

Step 1.1.2

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

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

Step 1.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 1.1.4

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

$0 + 2 x$

Step 1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 1.2

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

$\int {\sqrt{u_{1} - 1}}^{4} \sqrt{u_{1}} \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u_{1} - 1}}^{4}$ as ${(u_{1} - 1)}^{2}$ .

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Step 2.1.1

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

$\int {({(u_{1} - 1)}^{\frac{1}{2}})}^{4} \sqrt{u_{1}} \frac{1}{2} d u_{1}$

Step 2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int {(u_{1} - 1)}^{\frac{1}{2} \cdot 4} \sqrt{u_{1}} \frac{1}{2} d u_{1}$

Step 2.1.3

Combine $\frac{1}{2}$ and $4$ .

$\int {(u_{1} - 1)}^{\frac{4}{2}} \sqrt{u_{1}} \frac{1}{2} d u_{1}$

Step 2.1.4

Cancel the common factor of $4$ and $2$ .

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Step 2.1.4.1

Factor $2$ out of $4$ .

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

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $2$ out of $2$ .

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

Step 2.1.4.2.2

Cancel the common factor.

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

Step 2.1.4.2.3

Rewrite the expression.

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

Step 2.1.4.2.4

Divide $2$ by $1$ .

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

Step 2.2

Combine $\frac{1}{2}$ and ${(u_{1} - 1)}^{2}$ .

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

Step 2.3

Combine $\frac{{(u_{1} - 1)}^{2}}{2}$ and $\sqrt{u_{1}}$ .

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

Step 3

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

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

Step 4

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

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

Step 5

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 5.1

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

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Step 5.1.1

Differentiate $u_{1} - 1$ .

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

Step 5.1.2

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 5.1.3

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 5.1.4

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

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

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

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

Step 6

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 6.1

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

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Step 6.1.1

Differentiate $u_{2} + 1$ .

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

Step 6.1.2

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 6.1.3

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 6.1.4

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

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2