Calculus Examples

$\sec^{2} (\frac{θ}{2})$

Write $\sec^{2} (\frac{θ}{2})$ as a function.

$f (θ) = \sec^{2} (\frac{θ}{2})$

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

$F (θ) = \int f (θ) d θ$

Set up the integral to solve.

$F (θ) = \int \sec^{2} (\frac{θ}{2}) d θ$

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

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

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Differentiate $\frac{θ}{2}$ .

$\frac{d}{d θ} [\frac{θ}{2}]$

Since $\frac{1}{2}$ is constant with respect to $θ$ , the derivative of $\frac{θ}{2}$ with respect to $θ$ is $\frac{1}{2} \frac{d}{d θ} [θ]$ .

$\frac{1}{2} \frac{d}{d θ} [θ]$

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

$\frac{1}{2} \cdot 1$

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

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

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

Simplify.

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Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\int \sec^{2} (u) (1 \cdot 2) d u$

Multiply $2$ by $1$ .

$\int \sec^{2} (u) \cdot 2 d u$

Move $2$ to the left of $\sec^{2} (u)$ .

$\int 2 \sec^{2} (u) d u$

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int \sec^{2} (u) d u$

Since the derivative of $\tan (u)$ is $\sec^{2} (u)$ , the integral of $\sec^{2} (u)$ is $\tan (u)$ .

$2 (\tan (u) + C)$

Simplify.

$2 \tan (u) + C$

Replace all occurrences of $u$ with $\frac{θ}{2}$ .

$2 \tan (\frac{θ}{2}) + C$

Reorder terms.

$2 \tan (\frac{1}{2} θ) + C$

The answer is the antiderivative of the function $f (θ) = \sec^{2} (\frac{θ}{2})$ .

$F (θ) =$ $2 \tan (\frac{1}{2} θ) + C$