Calculus Examples

$9 x \cos (3 x)$

Step 1

Write $9 x \cos (3 x)$ as a function.

$f (x) = 9 x \cos (3 x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int 9 x \cos (3 x) d x$

Step 4

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int x \cos (3 x) d x$

Step 5

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

$9 (x (\frac{1}{3} \sin (3 x)) - \int \frac{1}{3} \sin (3 x) d x)$

Step 6

Simplify.

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Step 6.1

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

$9 (x \frac{\sin (3 x)}{3} - \int \frac{1}{3} \sin (3 x) d x)$

Step 6.2

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

$9 (\frac{x \sin (3 x)}{3} - \int \frac{1}{3} \sin (3 x) d x)$

Step 6.3

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

$9 (\frac{x \sin (3 x)}{3} - \int \frac{\sin (3 x)}{3} d x)$

Step 7

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

$9 (\frac{x \sin (3 x)}{3} - (\frac{1}{3} \int \sin (3 x) d x))$

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 $d u$ .

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

$9 (\frac{x \sin (3 x)}{3} - \frac{1}{3 \cdot 3} \int \sin (u) d u)$

Step 11.2

Multiply $3$ by $3$ .

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

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

Step 15.2

Cancel the common factor of $3$ .

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Step 15.2.1

Factor $3$ out of $9$ .

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

Step 15.2.2

Cancel the common factor.

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

Step 15.2.3

Rewrite the expression.

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

Step 15.3

Cancel the common factor of $9$ .

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Step 15.3.1

Cancel the common factor.

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

Step 15.3.2

Rewrite the expression.

$3 (x \sin (3 x)) + \cos (3 x) + C$

$3 x \sin (3 x) + \cos (3 x) + C$

Step 16

The answer is the antiderivative of the function $f (x) = 9 x \cos (3 x)$ .

$F (x) =$ $3 x \sin (3 x) + \cos (3 x) + C$