Calculus Examples

$\int (1 + \cos (3 x)) d x$

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $3 x$ .

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

Step 1.1.2

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 1.1.3

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 1.1.4

Multiply $3$ by $1$ .

$3$

Step 1.2

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

$\int (1 + \cos (u)) \frac{1}{3} d u$

Step 2

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

$\frac{1}{3} \int 1 + \cos (u) d u$

Step 3

Split the single integral into multiple integrals.

$\frac{1}{3} (\int d u + \int \cos (u) d u)$

Step 4

Apply the constant rule.

$\frac{1}{3} (u + C + \int \cos (u) d u)$

Step 5

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

$\frac{1}{3} (u + C + \sin (u) + C)$

Step 6

Simplify.

$\frac{1}{3} (u + \sin (u)) + C$

Step 7

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

$\frac{1}{3} (3 x + \sin (3 x)) + C$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{1}{3} (3 x) + \frac{1}{3} \sin (3 x) + C$

Step 8.2

Cancel the common factor of $3$ .

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Step 8.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) + \frac{1}{3} \sin (3 x) + C$

Step 8.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) + \frac{1}{3} \sin (3 x) + C$

Step 8.2.3

Rewrite the expression.

$x + \frac{1}{3} \sin (3 x) + C$

Step 8.3

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

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

Step 9

Reorder terms.

$x + \frac{1}{3} \sin (3 x) + C$