Calculus Examples

$\int 6 x^{5} \sin (x^{6}) d x$

Step 1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int x^{5} \sin (x^{6}) d x$

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $x^{2}$ .

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

Step 2.1.2

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 2.2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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Step 3.1.4.1

Factor $2$ out of $4$ .

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

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $2$ .

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

Step 3.1.4.2.2

Cancel the common factor.

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

Step 3.1.4.2.3

Rewrite the expression.

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

Step 3.1.4.2.4

Divide $2$ by $1$ .

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

Step 3.2

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

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Step 3.2.1

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

$6 \int {u_{1}}^{2} \sin ({({u_{1}}^{\frac{1}{2}})}^{6}) \frac{1}{2} d u_{1}$

Step 3.2.2

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

$6 \int {u_{1}}^{2} \sin ({u_{1}}^{\frac{1}{2} \cdot 6}) \frac{1}{2} d u_{1}$

Step 3.2.3

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

$6 \int {u_{1}}^{2} \sin ({u_{1}}^{\frac{6}{2}}) \frac{1}{2} d u_{1}$

Step 3.2.4

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

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Step 3.2.4.1

Factor $2$ out of $6$ .

$6 \int {u_{1}}^{2} \sin ({u_{1}}^{\frac{2 \cdot 3}{2}}) \frac{1}{2} d u_{1}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $2$ .

$6 \int {u_{1}}^{2} \sin ({u_{1}}^{\frac{2 \cdot 3}{2 (1)}}) \frac{1}{2} d u_{1}$

Step 3.2.4.2.2

Cancel the common factor.

$6 \int {u_{1}}^{2} \sin ({u_{1}}^{\frac{2 \cdot 3}{2 \cdot 1}}) \frac{1}{2} d u_{1}$

Step 3.2.4.2.3

Rewrite the expression.

$6 \int {u_{1}}^{2} \sin ({u_{1}}^{\frac{3}{1}}) \frac{1}{2} d u_{1}$

Step 3.2.4.2.4

Divide $3$ by $1$ .

$6 \int {u_{1}}^{2} \sin ({u_{1}}^{3}) \frac{1}{2} d u_{1}$

Step 3.3

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

$6 \int \frac{{u_{1}}^{2}}{2} \sin ({u_{1}}^{3}) d u_{1}$

Step 3.4

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

$6 \int \frac{{u_{1}}^{2} \sin ({u_{1}}^{3})}{2} d u_{1}$

Step 4

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

$6 (\frac{1}{2} \int {u_{1}}^{2} \sin ({u_{1}}^{3}) d u_{1})$

Step 5

Simplify.

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Step 5.1

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

$\frac{6}{2} \int {u_{1}}^{2} \sin ({u_{1}}^{3}) d u_{1}$

Step 5.2

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

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Step 5.2.1

Factor $2$ out of $6$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.2.2.4

Divide $3$ by $1$ .

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

Step 6

Let $u_{2} = {u_{1}}^{3}$ . Then $d u_{2} = 3 {u_{1}}^{2} d u_{1}$ , so $\frac{1}{3} d u_{2} = {u_{1}}^{2} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.1

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

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Step 6.1.1

Differentiate ${u_{1}}^{3}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{3}]$

Step 6.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$3 {u_{1}}^{2}$

Step 6.2

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

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

Step 7

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

$3 \int \frac{\sin (u_{2})}{3} d u_{2}$

Step 8

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

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

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{3}$ and $3$ .

$\frac{3}{3} \int \sin (u_{2}) d u_{2}$

Step 9.2

Cancel the common factor of $3$ .

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Step 9.2.1

Cancel the common factor.

$\frac{3}{3} \int \sin (u_{2}) d u_{2}$

Step 9.2.2

Rewrite the expression.

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

Step 9.3

Multiply $\int \sin (u_{2}) d u_{2}$ by $1$ .

$\int \sin (u_{2}) d u_{2}$

Step 10

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

$- \cos (u_{2}) + C$

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

$- \cos ({u_{1}}^{3}) + C$

Step 11.2

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

$- \cos ({(x^{2})}^{3}) + C$