Calculus Examples

$\int x \sqrt{1 - x^{4}} d x$

Step 1

Let $u_{1} = x^{2}$ . 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 $u_{1} = x^{2}$ . 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

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

$2 x$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

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 $4$ .

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

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $2$ out of $2$ .

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

Step 2.1.4.2.2

Cancel the common factor.

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

Step 2.1.4.2.3

Rewrite the expression.

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

Step 2.1.4.2.4

Divide $2$ by $1$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Let $u_{1} = \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u_{1} = \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\cos (t)$ is positive.

$\frac{1}{2} \int \sqrt{1 - \sin^{2} (t)} \cos (t) d t$

Step 5

Simplify terms.

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Step 5.1

Simplify $\sqrt{1 - \sin^{2} (t)}$ .

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Step 5.1.1

Apply pythagorean identity.

$\frac{1}{2} \int \sqrt{\cos^{2} (t)} \cos (t) d t$

Step 5.1.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{2} \int \cos (t) \cos (t) d t$

Step 5.2

Simplify.

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Step 5.2.1

Raise $\cos (t)$ to the power of $1$ .

$\frac{1}{2} \int \cos^{1} (t) \cos (t) d t$

Step 5.2.2

Raise $\cos (t)$ to the power of $1$ .

$\frac{1}{2} \int \cos^{1} (t) \cos^{1} (t) d t$

Step 5.2.3

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

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

Step 5.2.4

Add $1$ and $1$ .

$\frac{1}{2} \int \cos^{2} (t) d t$

Step 6

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$\frac{1}{2} \int \frac{1 + \cos (2 t)}{2} d t$

Step 7

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

$\frac{1}{2} (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2 \cdot 2} \int 1 + \cos (2 t) d t$

Step 8.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int 1 + \cos (2 t) d t$

Step 9

Split the single integral into multiple integrals.

$\frac{1}{4} (\int d t + \int \cos (2 t) d t)$

Step 10

Apply the constant rule.

$\frac{1}{4} (t + C + \int \cos (2 t) d t)$

Step 11

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

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Step 11.1

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

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Step 11.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 11.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 11.1.3

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

$2 \cdot 1$

Step 11.1.4

Multiply $2$ by $1$ .

$2$

Step 11.2

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

$\frac{1}{4} (t + C + \int \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 12

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

$\frac{1}{4} (t + C + \int \frac{\cos (u_{2})}{2} d u_{2})$

Step 13

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

$\frac{1}{4} (t + C + \frac{1}{2} \int \cos (u_{2}) d u_{2})$

Step 14

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{4} (t + C + \frac{1}{2} (\sin (u_{2}) + C))$

Step 15

Simplify.

$\frac{1}{4} (t + \frac{1}{2} \sin (u_{2})) + C$

Step 16

Substitute back in for each integration substitution variable.

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Step 16.1

Replace all occurrences of $t$ with $\arcsin (u_{1})$ .

$\frac{1}{4} (\arcsin (u_{1}) + \frac{1}{2} \sin (u_{2})) + C$

Step 16.2

Replace all occurrences of $u_{2}$ with $2 t$ .

$\frac{1}{4} (\arcsin (u_{1}) + \frac{1}{2} \sin (2 t)) + C$

Step 16.3

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

$\frac{1}{4} (\arcsin (x^{2}) + \frac{1}{2} \sin (2 t)) + C$

Step 16.4

Replace all occurrences of $t$ with $\arcsin (u_{1})$ .

$\frac{1}{4} (\arcsin (x^{2}) + \frac{1}{2} \sin (2 \arcsin (u_{1}))) + C$

Step 16.5

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

$\frac{1}{4} (\arcsin (x^{2}) + \frac{1}{2} \sin (2 \arcsin (x^{2}))) + C$

Step 17

Simplify.

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Step 17.1

Combine $\frac{1}{2}$ and $\sin (2 \arcsin (x^{2}))$ .

$\frac{1}{4} (\arcsin (x^{2}) + \frac{\sin (2 \arcsin (x^{2}))}{2}) + C$

Step 17.2

Apply the distributive property.

$\frac{1}{4} \arcsin (x^{2}) + \frac{1}{4} \cdot \frac{\sin (2 \arcsin (x^{2}))}{2} + C$

Step 17.3

Combine $\frac{1}{4}$ and $\arcsin (x^{2})$ .

$\frac{\arcsin (x^{2})}{4} + \frac{1}{4} \cdot \frac{\sin (2 \arcsin (x^{2}))}{2} + C$

Step 17.4

Combine.

$\frac{\arcsin (x^{2})}{4} + \frac{1 \sin (2 \arcsin (x^{2}))}{4 \cdot 2} + C$

Step 17.5

Simplify each term.

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Step 17.5.1

Multiply $\sin (2 \arcsin (x^{2}))$ by $1$ .

$\frac{\arcsin (x^{2})}{4} + \frac{\sin (2 \arcsin (x^{2}))}{4 \cdot 2} + C$

Step 17.5.2

Multiply $4$ by $2$ .

$\frac{\arcsin (x^{2})}{4} + \frac{\sin (2 \arcsin (x^{2}))}{8} + C$

Step 18

Reorder terms.

$\frac{1}{4} \arcsin (x^{2}) + \frac{1}{8} \sin (2 \arcsin (x^{2})) + C$