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Question
\[\int x \sec^2 2x\ dx\]
Sum
Solution
\[\int x_I \cdot \sec^2 2_{II}x\ dx \]
\[ = x\int \sec^2 2x\ dx - \int\left\{ \frac{d}{dx}\left( x \right)\int \sec^2 2x\ dx \right\}dx\]
\[ = \frac{x \tan 2x}{2} - \int1 \cdot \frac{\tan 2x}{2} dx\]
\[ = \frac{x \tan 2x}{2} - \frac{1}{2} \frac{\text{ ln } \left| \sec 2x \right|}{2} + C\]
\[ = \frac{x \tan 2x}{2} - \frac{1}{4} \text{ ln} \left| \sec 2x \right| + C\]
\[ = x\int \sec^2 2x\ dx - \int\left\{ \frac{d}{dx}\left( x \right)\int \sec^2 2x\ dx \right\}dx\]
\[ = \frac{x \tan 2x}{2} - \int1 \cdot \frac{\tan 2x}{2} dx\]
\[ = \frac{x \tan 2x}{2} - \frac{1}{2} \frac{\text{ ln } \left| \sec 2x \right|}{2} + C\]
\[ = \frac{x \tan 2x}{2} - \frac{1}{4} \text{ ln} \left| \sec 2x \right| + C\]
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