Calculus Examples

$\frac{x}{6} \tan^{2} (\frac{x}{6})$

Step 1

Combine $\frac{x}{6}$ and $\tan^{2} (\frac{x}{6})$ .

$\int \frac{x \tan^{2} (\frac{x}{6})}{6} d x$

Step 2

Since $\frac{1}{6}$ is constant with respect to $x$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int x \tan^{2} (\frac{x}{6}) d x$

Step 3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \tan^{2} (\frac{x}{6})$ .

$\frac{1}{6} (x (- x + 6 \tan (\frac{1}{6} x)) - \int - x + 6 \tan (\frac{1}{6} x) d x)$

Step 4

Split the single integral into multiple integrals.

$\frac{1}{6} (x (- x + 6 \tan (\frac{1}{6} x)) - (\int - x d x + \int 6 \tan (\frac{1}{6} x) d x))$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{6} (x (- x + 6 \tan (\frac{1}{6} x)) - (- \int x d x + \int 6 \tan (\frac{1}{6} x) d x))$

Step 6

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{6} (x (- x + 6 \tan (\frac{1}{6} x)) - (- (\frac{1}{2} x^{2} + C) + \int 6 \tan (\frac{1}{6} x) d x))$

Step 7

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

$\frac{1}{6} (x (- x + 6 \tan (\frac{1}{6} x)) - (- (\frac{1}{2} x^{2} + C) + 6 \int \tan (\frac{1}{6} x) d x))$

Step 8

Let $u = \frac{1}{6} x$ . Then $d u = \frac{1}{6} d x$ , so $6 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = \frac{1}{6} x$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $\frac{1}{6} x$ .

$\frac{d}{d x} [\frac{1}{6} x]$

Step 8.1.2

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{1}{6} x$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x]$ .

$\frac{1}{6} \frac{d}{d x} [x]$

Step 8.1.3

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

$\frac{1}{6} \cdot 1$

Step 8.1.4

Multiply $\frac{1}{6}$ by $1$ .

$\frac{1}{6}$

Step 8.2

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{6} (x (- x + 6 \tan (\frac{1}{6} x)) - (- (\frac{1}{2} x^{2} + C) + 6 \int \tan (u) \frac{1}{\frac{1}{6}} d u))$

Step 9

Simplify.

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Step 9.1

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

$\frac{1}{6} (x (- x + 6 \tan (\frac{x}{6})) - (- (\frac{1}{2} x^{2} + C) + 6 \int \tan (u) \frac{1}{\frac{1}{6}} d u))$

Step 9.2

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

$\frac{1}{6} (x (- x + 6 \tan (\frac{x}{6})) - (- (\frac{x^{2}}{2} + C) + 6 \int \tan (u) \frac{1}{\frac{1}{6}} d u))$

Step 9.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{6}$ .

$\frac{1}{6} (x (- x + 6 \tan (\frac{x}{6})) - (- (\frac{x^{2}}{2} + C) + 6 \int \tan (u) (1 \cdot 6) d u))$

Step 9.4

Multiply $6$ by $1$ .

$\frac{1}{6} (x (- x + 6 \tan (\frac{x}{6})) - (- (\frac{x^{2}}{2} + C) + 6 \int \tan (u) \cdot 6 d u))$

Step 9.5

Move $6$ to the left of $\tan (u)$ .

$\frac{1}{6} (x (- x + 6 \tan (\frac{x}{6})) - (- (\frac{x^{2}}{2} + C) + 6 \int 6 \tan (u) d u))$

Step 10

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

$\frac{1}{6} (x (- x + 6 \tan (\frac{x}{6})) - (- (\frac{x^{2}}{2} + C) + 6 (6 \int \tan (u) d u)))$

Step 11

Multiply $6$ by $6$ .

$\frac{1}{6} (x (- x + 6 \tan (\frac{x}{6})) - (- (\frac{x^{2}}{2} + C) + 36 \int \tan (u) d u))$

Step 12

The integral of $\tan (u)$ with respect to $u$ is $\ln (| \sec (u) |)$ .

$\frac{1}{6} (x (- x + 6 \tan (\frac{x}{6})) - (- (\frac{x^{2}}{2} + C) + 36 (\ln (| \sec (u) |) + C)))$

Step 13

Simplify.

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Step 13.1

Simplify.

$\frac{1}{6} (- x^{2} + 6 x \tan (\frac{x}{6}) + \frac{x^{2}}{2} - 36 \ln (| \sec (u) |)) + C$

Step 13.2

Simplify.

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Step 13.2.1

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{6} (6 x \tan (\frac{x}{6}) - x^{2} \cdot \frac{2}{2} + \frac{x^{2}}{2} - 36 \ln (| \sec (u) |)) + C$

Step 13.2.2

Combine $- x^{2}$ and $\frac{2}{2}$ .

$\frac{1}{6} (6 x \tan (\frac{x}{6}) + \frac{- x^{2} \cdot 2}{2} + \frac{x^{2}}{2} - 36 \ln (| \sec (u) |)) + C$

Step 13.2.3

Combine the numerators over the common denominator.

$\frac{1}{6} (6 x \tan (\frac{x}{6}) + \frac{- x^{2} \cdot 2 + x^{2}}{2} - 36 \ln (| \sec (u) |)) + C$

Step 13.2.4

Multiply $2$ by $- 1$ .

$\frac{1}{6} (6 x \tan (\frac{x}{6}) + \frac{- 2 x^{2} + x^{2}}{2} - 36 \ln (| \sec (u) |)) + C$

Step 13.2.5

Add $- 2 x^{2}$ and $x^{2}$ .

$\frac{1}{6} (6 x \tan (\frac{x}{6}) + \frac{- x^{2}}{2} - 36 \ln (| \sec (u) |)) + C$

Step 13.2.6

Move the negative in front of the fraction.

$\frac{1}{6} (6 x \tan (\frac{x}{6}) - \frac{x^{2}}{2} - 36 \ln (| \sec (u) |)) + C$

Step 14

Replace all occurrences of $u$ with $\frac{1}{6} x$ .

$\frac{1}{6} (6 x \tan (\frac{x}{6}) - \frac{x^{2}}{2} - 36 \ln (| \sec (\frac{1}{6} x) |)) + C$

Step 15

Simplify.

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Step 15.1

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

$\frac{1}{6} (6 x \tan (\frac{x}{6}) - \frac{x^{2}}{2} - 36 \ln (| \sec (\frac{x}{6}) |)) + C$

Step 15.2

Apply the distributive property.

$\frac{1}{6} (6 x \tan (\frac{x}{6})) + \frac{1}{6} (- \frac{x^{2}}{2}) + \frac{1}{6} (- 36 \ln (| \sec (\frac{x}{6}) |)) + C$

Step 15.3

Simplify.

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Step 15.3.1

Cancel the common factor of $6$ .

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Step 15.3.1.1

Factor $6$ out of $6 x \tan (\frac{x}{6})$ .

$\frac{1}{6} (6 (x \tan (\frac{x}{6}))) + \frac{1}{6} (- \frac{x^{2}}{2}) + \frac{1}{6} (- 36 \ln (| \sec (\frac{x}{6}) |)) + C$

Step 15.3.1.2

Cancel the common factor.

$\frac{1}{6} (6 (x \tan (\frac{x}{6}))) + \frac{1}{6} (- \frac{x^{2}}{2}) + \frac{1}{6} (- 36 \ln (| \sec (\frac{x}{6}) |)) + C$

Step 15.3.1.3

Rewrite the expression.

$x \tan (\frac{x}{6}) + \frac{1}{6} (- \frac{x^{2}}{2}) + \frac{1}{6} (- 36 \ln (| \sec (\frac{x}{6}) |)) + C$

Step 15.3.2

Multiply $\frac{1}{6} (- \frac{x^{2}}{2})$ .

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Step 15.3.2.1

Multiply $\frac{1}{6}$ by $\frac{x^{2}}{2}$ .

$x \tan (\frac{x}{6}) - \frac{x^{2}}{6 \cdot 2} + \frac{1}{6} (- 36 \ln (| \sec (\frac{x}{6}) |)) + C$

Step 15.3.2.2

Multiply $6$ by $2$ .

$x \tan (\frac{x}{6}) - \frac{x^{2}}{12} + \frac{1}{6} (- 36 \ln (| \sec (\frac{x}{6}) |)) + C$

Step 15.3.3

Cancel the common factor of $6$ .

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Step 15.3.3.1

Factor $6$ out of $- 36 \ln (| \sec (\frac{x}{6}) |)$ .

$x \tan (\frac{x}{6}) - \frac{x^{2}}{12} + \frac{1}{6} (6 (- 6 \ln (| \sec (\frac{x}{6}) |))) + C$

Step 15.3.3.2

Cancel the common factor.

$x \tan (\frac{x}{6}) - \frac{x^{2}}{12} + \frac{1}{6} (6 (- 6 \ln (| \sec (\frac{x}{6}) |))) + C$

Step 15.3.3.3

Rewrite the expression.

$x \tan (\frac{x}{6}) - \frac{x^{2}}{12} - 6 \ln (| \sec (\frac{x}{6}) |) + C$

Step 16

Reorder terms.

$x \tan (\frac{1}{6} x) - \frac{1}{12} x^{2} - 6 \ln (| \sec (\frac{1}{6} x) |) + C$