Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \sin (3 x)$ .

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

Step 2

Simplify.

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Step 2.1

Combine $\cos (3 x)$ and $\frac{1}{3}$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $2$ by $- 1$ .

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

Step 4.2

Combine $- 2$ and $\frac{1}{3}$ .

$- \frac{x^{2} \cos (3 x)}{3} - (\frac{- 2}{3} \int \cos (3 x) (x) d x)$

Step 4.3

Move the negative in front of the fraction.

$- \frac{x^{2} \cos (3 x)}{3} - (- \frac{2}{3} \int \cos (3 x) (x) d x)$

Step 4.4

Multiply $- 1$ by $- 1$ .

$- \frac{x^{2} \cos (3 x)}{3} + 1 (\frac{2}{3} \int \cos (3 x) (x) d x)$

Step 4.5

Multiply $\frac{2}{3}$ by $1$ .

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} \int \cos (3 x) (x) d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \cos (3 x)$ .

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (x (\frac{1}{3} \sin (3 x)) - \int \frac{1}{3} \sin (3 x) d x)$

Step 6

Simplify.

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Step 6.1

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

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (x \frac{\sin (3 x)}{3} - \int \frac{1}{3} \sin (3 x) d x)$

Step 6.2

Combine $x$ and $\frac{\sin (3 x)}{3}$ .

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \int \frac{1}{3} \sin (3 x) d x)$

Step 6.3

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

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \int \frac{\sin (3 x)}{3} d x)$

Step 7

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

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - (\frac{1}{3} \int \sin (3 x) d x))$

Step 8

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 8.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 8.1.3

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

$3 \cdot 1$

Step 8.1.4

Multiply $3$ by $1$ .

$3$

Step 8.2

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

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \frac{1}{3} \int \sin (u) \frac{1}{3} d u)$

Step 9

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

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \frac{1}{3} \int \frac{\sin (u)}{3} d u)$

Step 10

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

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \frac{1}{3} (\frac{1}{3} \int \sin (u) d u))$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \frac{1}{3 \cdot 3} \int \sin (u) d u)$

Step 11.2

Multiply $3$ by $3$ .

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \frac{1}{9} \int \sin (u) d u)$

Step 12

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \frac{1}{9} (- \cos (u) + C))$

Step 13

Simplify.

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Step 13.1

Rewrite $- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \frac{1}{9} (- \cos (u) + C))$ as $- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9}) + C$ .

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9}) + C$

Step 13.2

Simplify.

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Step 13.2.1

To write $\frac{2}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9}) \cdot \frac{3}{3} + C$

Step 13.2.2

Combine $\frac{2}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9})$ and $\frac{3}{3}$ .

$- \frac{x^{2} \cos (3 x)}{3} + \frac{\frac{2}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9}) \cdot 3}{3} + C$

Step 13.2.3

Combine the numerators over the common denominator.

$\frac{- x^{2} \cos (3 x) + \frac{2}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9}) \cdot 3}{3} + C$

Step 13.2.4

Combine $3$ and $\frac{2}{3}$ .

$\frac{- x^{2} \cos (3 x) + \frac{3 \cdot 2}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9})}{3} + C$

Step 13.2.5

Multiply $3$ by $2$ .

$\frac{- x^{2} \cos (3 x) + \frac{6}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9})}{3} + C$

Step 13.2.6

Cancel the common factor of $6$ and $3$ .

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Step 13.2.6.1

Factor $3$ out of $6$ .

$\frac{- x^{2} \cos (3 x) + \frac{3 \cdot 2}{3} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9})}{3} + C$

Step 13.2.6.2

Cancel the common factors.

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Step 13.2.6.2.1

Factor $3$ out of $3$ .

$\frac{- x^{2} \cos (3 x) + \frac{3 \cdot 2}{3 (1)} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9})}{3} + C$

Step 13.2.6.2.2

Cancel the common factor.

$\frac{- x^{2} \cos (3 x) + \frac{3 \cdot 2}{3 \cdot 1} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9})}{3} + C$

Step 13.2.6.2.3

Rewrite the expression.

$\frac{- x^{2} \cos (3 x) + \frac{2}{1} (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9})}{3} + C$

Step 13.2.6.2.4

Divide $2$ by $1$ .

$\frac{- x^{2} \cos (3 x) + 2 (\frac{x \sin (3 x)}{3} + \frac{\cos (u)}{9})}{3} + C$

Step 14

Replace all occurrences of $u$ with $3 x$ .

$\frac{- x^{2} \cos (3 x) + 2 (\frac{x \sin (3 x)}{3} + \frac{\cos (3 x)}{9})}{3} + C$

Step 15

Simplify.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Apply the distributive property.

$\frac{- x^{2} \cos (3 x) + 2 \frac{x \sin (3 x)}{3} + 2 \frac{\cos (3 x)}{9}}{3} + C$

Step 15.1.2

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

$\frac{- x^{2} \cos (3 x) + \frac{2 (x \sin (3 x))}{3} + 2 \frac{\cos (3 x)}{9}}{3} + C$

Step 15.1.3

Combine $2$ and $\frac{\cos (3 x)}{9}$ .

$\frac{- x^{2} \cos (3 x) + \frac{2 (x \sin (3 x))}{3} + \frac{2 \cos (3 x)}{9}}{3} + C$

Step 15.1.4

To write $- x^{2} \cos (3 x)$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{2 x \sin (3 x)}{3} - x^{2} \cos (3 x) \cdot \frac{9}{9} + \frac{2 \cos (3 x)}{9}}{3} + C$

Step 15.1.5

Combine $- x^{2} \cos (3 x)$ and $\frac{9}{9}$ .

$\frac{\frac{2 x \sin (3 x)}{3} + \frac{- x^{2} \cos (3 x) \cdot 9}{9} + \frac{2 \cos (3 x)}{9}}{3} + C$

Step 15.1.6

Combine the numerators over the common denominator.

$\frac{\frac{2 x \sin (3 x)}{3} + \frac{- x^{2} \cos (3 x) \cdot 9 + 2 \cos (3 x)}{9}}{3} + C$

Step 15.1.7

To write $\frac{2 x \sin (3 x)}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{2 x \sin (3 x)}{3} \cdot \frac{3}{3} + \frac{- x^{2} \cos (3 x) \cdot 9 + 2 \cos (3 x)}{9}}{3} + C$

Step 15.1.8

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.1.8.1

Multiply $\frac{2 x \sin (3 x)}{3}$ by $\frac{3}{3}$ .

$\frac{\frac{2 x \sin (3 x) \cdot 3}{3 \cdot 3} + \frac{- x^{2} \cos (3 x) \cdot 9 + 2 \cos (3 x)}{9}}{3} + C$

Step 15.1.8.2

Multiply $3$ by $3$ .

$\frac{\frac{2 x \sin (3 x) \cdot 3}{9} + \frac{- x^{2} \cos (3 x) \cdot 9 + 2 \cos (3 x)}{9}}{3} + C$

Step 15.1.9

Combine the numerators over the common denominator.

$\frac{\frac{2 x \sin (3 x) \cdot 3 - x^{2} \cos (3 x) \cdot 9 + 2 \cos (3 x)}{9}}{3} + C$

Step 15.1.10

Rewrite $\frac{2 x \sin (3 x) \cdot 3 - x^{2} \cos (3 x) \cdot 9 + 2 \cos (3 x)}{9}$ in a factored form.

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Step 15.1.10.1

Multiply $3$ by $2$ .

$\frac{\frac{6 x \sin (3 x) - x^{2} \cos (3 x) \cdot 9 + 2 \cos (3 x)}{9}}{3} + C$

Step 15.1.10.2

Multiply $9$ by $- 1$ .

$\frac{\frac{6 x \sin (3 x) - 9 x^{2} \cos (3 x) + 2 \cos (3 x)}{9}}{3} + C$

Step 15.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{6 x \sin (3 x) - 9 x^{2} \cos (3 x) + 2 \cos (3 x)}{9} \cdot \frac{1}{3} + C$

Step 15.3

Multiply $\frac{6 x \sin (3 x) - 9 x^{2} \cos (3 x) + 2 \cos (3 x)}{9} \cdot \frac{1}{3}$ .

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Step 15.3.1

Multiply $\frac{6 x \sin (3 x) - 9 x^{2} \cos (3 x) + 2 \cos (3 x)}{9}$ by $\frac{1}{3}$ .

$\frac{6 x \sin (3 x) - 9 x^{2} \cos (3 x) + 2 \cos (3 x)}{9 \cdot 3} + C$

Step 15.3.2

Multiply $9$ by $3$ .

$\frac{6 x \sin (3 x) - 9 x^{2} \cos (3 x) + 2 \cos (3 x)}{27} + C$

Step 15.4

Reorder terms.

$\frac{1}{27} (6 x \sin (3 x) - 9 x^{2} \cos (3 x) + 2 \cos (3 x)) + C$