Question

Determine the
∫sec2x
with respect to x.

Answer 1

Answer on Question #45728 – Math - Integral Calculus

Determine the $\int \sec 2x$ with respect to $x$.

Solution

$$I = \int \sec 2x \, dx = \int \frac{\sec 2x(\sec 2x + \tan 2x)}{\sec 2x + \tan 2x} \, dx$$

Substitute $u = \sec 2x + \tan 2x \rightarrow du = (2 \tan 2x \sec 2x + 2 \sec^2 2x) \, dx = 2 \sec 2x(\tan 2x + \sec 2x) \, dx$

So, $I = \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln |u| + Const = \frac{1}{2} \ln |\sec 2x + \tan 2x| + Const = \frac{1}{2} \ln \left| \frac{1 + \sin 2x}{\cos 2x} \right| + Const = \frac{1}{2} \ln |1 + \sin 2x| - \frac{1}{2} \ln |\cos 2x| + Const.$

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