Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

Multiply $\int \sin (3 x) (x^{2}) d x$ by $1$ .

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

Step 5

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

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

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Multiply $2$ by $- 1$ .

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

Simplify.

Tap for more steps...

Step 8.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2

Multiply $\int \frac{2 \cos (3 x) x}{3} d x$ by $1$ .

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

Tap for more steps...

Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

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

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

Step 13

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

Tap for more steps...

Step 13.1

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

Tap for more steps...

Step 13.1.1

Differentiate $3 x$ .

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

Step 13.1.2

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 13.1.3

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 13.1.4

Multiply $3$ by $1$ .

$3$

Step 13.2

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

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify.

Tap for more steps...

Step 16.1

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

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

Step 16.2

Multiply $3$ by $3$ .

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

Step 17

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

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

Step 18

Simplify.

Tap for more steps...

Step 18.1

Rewrite $\frac{x^{3} \sin (3 x)}{3} - (- \frac{x^{2} \cos (3 x)}{3} + \frac{2}{3} (\frac{x \sin (3 x)}{3} - \frac{1}{9} (- \cos (u) + C)))$ as $\frac{x^{3} \sin (3 x)}{3} - (- \frac{x^{2} \cos (3 x)}{3} + \frac{2 x \sin (3 x)}{9} + \frac{2 \cos (u)}{27}) + C$ .

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

Step 18.2

Simplify.

Tap for more steps...

Step 18.2.1

To write $- (- \frac{x^{2} \cos (3 x)}{3} + \frac{2 x \sin (3 x)}{9} + \frac{2 \cos (u)}{27})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 18.2.2

Combine $- (- \frac{x^{2} \cos (3 x)}{3} + \frac{2 x \sin (3 x)}{9} + \frac{2 \cos (u)}{27})$ and $\frac{3}{3}$ .

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

Step 18.2.3

Combine the numerators over the common denominator.

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

Step 18.2.4

Multiply $3$ by $- 1$ .

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

Step 19

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

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

Step 20

Simplify.

Tap for more steps...

Step 20.1

Apply the distributive property.

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

Step 20.2

Simplify.

Tap for more steps...

Step 20.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 20.2.1.1

Move the leading negative in $- \frac{x^{2} \cos (3 x)}{3}$ into the numerator.

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

Step 20.2.1.2

Factor $3$ out of $- 3$ .

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

Step 20.2.1.3

Cancel the common factor.

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

Step 20.2.1.4

Rewrite the expression.

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

Step 20.2.2

Multiply $- 1$ by $- 1$ .

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

Step 20.2.3

Multiply $x^{2}$ by $1$ .

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

Step 20.2.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 20.2.4.1

Factor $3$ out of $- 3$ .

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

Step 20.2.4.2

Factor $3$ out of $9$ .

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

Step 20.2.4.3

Cancel the common factor.

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

Step 20.2.4.4

Rewrite the expression.

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

Step 20.2.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 20.2.5.1

Factor $3$ out of $- 3$ .

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

Step 20.2.5.2

Factor $3$ out of $27$ .

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

Step 20.2.5.3

Cancel the common factor.

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

Step 20.2.5.4

Rewrite the expression.

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

Step 20.3

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

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

Step 20.4

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

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

Step 20.5

Combine the numerators over the common denominator.

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

Step 20.6

Simplify the numerator.

Tap for more steps...

Step 20.6.1

Factor $\cos (3 x)$ out of $x^{2} \cos (3 x) \cdot 9 - 2 \cos (3 x)$ .

Tap for more steps...

Step 20.6.1.1

Factor $\cos (3 x)$ out of $x^{2} \cos (3 x) \cdot 9$ .

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

Step 20.6.1.2

Factor $\cos (3 x)$ out of $- 2 \cos (3 x)$ .

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

Step 20.6.1.3

Factor $\cos (3 x)$ out of $\cos (3 x) (x^{2} \cdot 9) + \cos (3 x) \cdot - 2$ .

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

Step 20.6.2

Move $9$ to the left of $x^{2}$ .

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

Step 20.7

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

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

Step 20.8

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

Tap for more steps...

Step 20.8.1

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

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

Step 20.8.2

Multiply $3$ by $3$ .

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

Step 20.9

Combine the numerators over the common denominator.

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

Step 20.10

Simplify the numerator.

Tap for more steps...

Step 20.10.1

Multiply $3$ by $- 2$ .

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

Step 20.10.2

Apply the distributive property.

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

Step 20.10.3

Rewrite using the commutative property of multiplication.

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

Step 20.10.4

Move $- 2$ to the left of $\cos (3 x)$ .

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

Step 20.11

Factor $- 1$ out of $- 6 x \sin (3 x)$ .

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

Step 20.12

Factor $- 1$ out of $9 \cos (3 x) x^{2}$ .

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

Step 20.13

Factor $- 1$ out of $- (6 x \sin (3 x)) - (- 9 \cos (3 x) x^{2})$ .

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

Step 20.14

Factor $- 1$ out of $- 2 \cos (3 x)$ .

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

Step 20.15

Factor $- 1$ out of $- (6 x \sin (3 x) - 9 \cos (3 x) x^{2}) - (2 \cos (3 x))$ .

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

Step 20.16

Rewrite $- (6 x \sin (3 x) - 9 \cos (3 x) x^{2} + 2 \cos (3 x))$ as $- 1 (6 x \sin (3 x) - 9 \cos (3 x) x^{2} + 2 \cos (3 x))$ .

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

Step 20.17

Move the negative in front of the fraction.

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

Step 20.18

Reorder factors in $- \frac{6 x \sin (3 x) - 9 \cos (3 x) x^{2} + 2 \cos (3 x)}{9}$ .

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

Step 21

Simplify.

Tap for more steps...

Step 21.1

Simplify the numerator.

Tap for more steps...

Step 21.1.1

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

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

Step 21.1.2

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

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

Step 21.1.3

Combine the numerators over the common denominator.

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

Step 21.1.4

Rewrite $\frac{x^{3} \sin (3 x) \cdot 9 - (6 x \sin (3 x) - 9 x^{2} \cos (3 x) + 2 \cos (3 x))}{9}$ in a factored form.

Tap for more steps...

Step 21.1.4.1

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

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

Step 21.1.4.2

Apply the distributive property.

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

Step 21.1.4.3

Simplify.

Tap for more steps...

Step 21.1.4.3.1

Multiply $6$ by $- 1$ .

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

Step 21.1.4.3.2

Multiply $- 9$ by $- 1$ .

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

Step 21.1.4.3.3

Multiply $2$ by $- 1$ .

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

Step 21.1.4.4

Simplify each term.

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

Step 21.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 21.3

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

Tap for more steps...

Step 21.3.1

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

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

Step 21.3.2

Multiply $9$ by $3$ .

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

Step 21.4

Reorder terms.

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