Calculus Examples

$\int x^{7} \cos (x^{4}) d x$

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 $d u_{1}$ .

$\int {\sqrt{u_{1}}}^{6} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u_{1}}}^{6}$ as ${u_{1}}^{3}$ .

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Step 2.1.1

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

$\int {({u_{1}}^{\frac{1}{2}})}^{6} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2.1.2

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

$\int {u_{1}}^{\frac{1}{2} \cdot 6} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2.1.3

Combine $\frac{1}{2}$ and $6$ .

$\int {u_{1}}^{\frac{6}{2}} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2.1.4

Cancel the common factor of $6$ and $2$ .

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Step 2.1.4.1

Factor $2$ out of $6$ .

$\int {u_{1}}^{\frac{2 \cdot 3}{2}} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $2$ out of $2$ .

$\int {u_{1}}^{\frac{2 \cdot 3}{2 (1)}} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2.1.4.2.2

Cancel the common factor.

$\int {u_{1}}^{\frac{2 \cdot 3}{2 \cdot 1}} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2.1.4.2.3

Rewrite the expression.

$\int {u_{1}}^{\frac{3}{1}} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2.1.4.2.4

Divide $3$ by $1$ .

$\int {u_{1}}^{3} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2.2

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

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Step 2.2.1

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

$\int {u_{1}}^{3} \cos ({({u_{1}}^{\frac{1}{2}})}^{4}) \frac{1}{2} d u_{1}$

Step 2.2.2

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

$\int {u_{1}}^{3} \cos ({u_{1}}^{\frac{1}{2} \cdot 4}) \frac{1}{2} d u_{1}$

Step 2.2.3

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

$\int {u_{1}}^{3} \cos ({u_{1}}^{\frac{4}{2}}) \frac{1}{2} d u_{1}$

Step 2.2.4

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

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Step 2.2.4.1

Factor $2$ out of $4$ .

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

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $2$ out of $2$ .

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

Step 2.2.4.2.2

Cancel the common factor.

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

Step 2.2.4.2.3

Rewrite the expression.

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

Step 2.2.4.2.4

Divide $2$ by $1$ .

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

Step 2.3

Combine $\frac{1}{2}$ and ${u_{1}}^{3}$ .

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

Step 2.4

Combine $\frac{{u_{1}}^{3}}{2}$ and $\cos ({u_{1}}^{2})$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate ${u_{1}}^{2}$ .

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

Step 4.1.2

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

$2 u_{1}$

Step 4.2

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Multiply $2$ by $2$ .

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

Step 8

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u_{2}$ and $dv = \cos (u_{2})$ .

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

Step 9

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

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

Step 10

Rewrite $\frac{1}{4} (u_{2} \sin (u_{2}) - (- \cos (u_{2}) + C))$ as $\frac{1}{4} (u_{2} \sin (u_{2}) + \cos (u_{2})) + C$ .

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

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

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

Step 11.2

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

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

Step 12

Simplify.

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Step 12.1

Simplify each term.

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Step 12.1.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 12.1.1.1

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

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

Step 12.1.1.2

Multiply $2$ by $2$ .

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

Step 12.1.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 12.1.2.1

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

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

Step 12.1.2.2

Multiply $2$ by $2$ .

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

Step 12.1.3

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 12.1.3.1

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

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

Step 12.1.3.2

Multiply $2$ by $2$ .

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Multiply $\frac{1}{4} (x^{4} \sin (x^{4}))$ .

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Step 12.3.1

Combine $x^{4}$ and $\frac{1}{4}$ .

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

Step 12.3.2

Combine $\frac{x^{4}}{4}$ and $\sin (x^{4})$ .

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

Step 12.4

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

$\frac{x^{4} \sin (x^{4})}{4} + \frac{\cos (x^{4})}{4} + C$

Step 13

Reorder terms.

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