Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2

Multiply $\int \frac{\cos (3 x)}{3} d x$ by $1$ .

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

Step 9

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

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

Step 10

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 10.1

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

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Step 10.1.1

Differentiate $3 x$ .

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

Step 10.1.2

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 10.1.3

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 10.1.4

Multiply $3$ by $1$ .

$3$

Step 10.2

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

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Multiply $3$ by $3$ .

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

Step 14

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

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

Step 15

Simplify.

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Step 15.1

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

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

Step 15.2

Simplify.

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Step 15.2.1

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

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

Step 15.2.2

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

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

Step 15.2.3

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

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

Step 15.2.4

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

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

Step 15.2.5

Combine $\frac{1}{9}$ and $\sin (u)$ .

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

Step 15.2.6

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

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

Step 15.2.7

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

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

Step 15.2.8

Combine the numerators over the common denominator.

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

Step 15.2.9

Multiply $3$ by $- 1$ .

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

Step 15.2.10

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

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

Step 15.2.11

Multiply $- 3$ by $2$ .

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

Step 15.2.12

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

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Step 15.2.12.1

Factor $3$ out of $- 6$ .

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

Step 15.2.12.2

Cancel the common factors.

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Step 15.2.12.2.1

Factor $3$ out of $3$ .

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

Step 15.2.12.2.2

Cancel the common factor.

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

Step 15.2.12.2.3

Rewrite the expression.

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

Step 15.2.12.2.4

Divide $- 2$ by $1$ .

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

Step 16

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

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

Step 17

Simplify.

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Step 17.1

Reorder factors in $\frac{\cos (3 x) x}{3}$ .

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

Step 17.2

Simplify the numerator.

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Step 17.2.1

Apply the distributive property.

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

Step 17.2.2

Multiply $- 2 (- \frac{x \cos (3 x)}{3})$ .

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Step 17.2.2.1

Multiply $- 1$ by $- 2$ .

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

Step 17.2.2.2

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

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

Step 17.2.3

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

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

Step 17.2.4

Move the negative in front of the fraction.

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

Step 17.2.5

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

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

Step 17.2.6

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

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

Step 17.2.7

Combine the numerators over the common denominator.

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

Step 17.2.8

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

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

Step 17.2.9

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

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Step 17.2.9.1

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

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

Step 17.2.9.2

Multiply $3$ by $3$ .

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

Step 17.2.10

Combine the numerators over the common denominator.

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

Step 17.2.11

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

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Step 17.2.11.1

Multiply $3$ by $2$ .

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

Step 17.2.11.2

Move $9$ to the left of $x^{2} \sin (3 x)$ .

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

Step 17.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 17.4

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

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Step 17.4.1

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

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

Step 17.4.2

Multiply $9$ by $3$ .

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

Step 17.5

Reorder terms.

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