Find the Integral sec(2x)^2

$\sec^{2} (2 x)$

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

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

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Rewrite.

$\frac{2}{1}$

Divide $2$ by $1$ .

$2$

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

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

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

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

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

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

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

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

Simplify.

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

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

$\frac{1}{2} \tan (2 x) + C$

Find the Integral sec(2x)^2

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