Calculus Examples

$\cos (2 θ)$

Step 1

Let $u = 2 θ$ . Then $d u = 2 d θ$ , so $\frac{1}{2} d u = d θ$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 θ$ . Find $\frac{d u}{d θ}$ .

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Step 1.1.1

Differentiate $2 θ$ .

$\frac{d}{d θ} [2 θ]$

Step 1.1.2

Since $2$ is constant with respect to $θ$ , the derivative of $2 θ$ with respect to $θ$ is $2 \frac{d}{d θ} [θ]$ .

$2 \frac{d}{d θ} [θ]$

Step 1.1.3

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

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

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

$\int \cos (u) \frac{1}{2} d u$

Step 2

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\int \frac{\cos (u)}{2} d u$

Step 3

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

$\frac{1}{2} \int \cos (u) d u$

Step 4

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

$\frac{1}{2} (\sin (u) + C)$

Step 5

Simplify.

$\frac{1}{2} \sin (u) + C$

Step 6

Replace all occurrences of $u$ with $2 θ$ .

$\frac{1}{2} \sin (2 θ) + C$