Question

Решить неопределенный интеграл

Answer 1

Ответ:

$\int\limits { \sec}^{4} (x) { \tg }^{2} (x)dx = \int\limits \frac{1}{ { \cos }^{4} (x)} { \tg }^{4} (x)dx = \\ = \int\limits\frac{ { \tg}^{4} (x)}{ { \cos}^{2}(x) } \times \frac{1}{ { \cos}^{2} (x)} dx = \\ = \int\limits \frac{ { \tg }^{4} (x)}{ { \cos }^{2}(x) } d( \tg(x)) \\ \\ \frac{1}{ { \cos }^{2}(x) } = 1 + { \tg }^{2} (x) \\ \\ \int\limits { \tg}^{4} (x) \times (1 + { \tg}^{2} (x))d (\tg(x) ) = \\ = \int\limits { \tg }^{6} (x)d( \tg(x)) + \int\limits { \tg}^{4} (x)d (\tg(x)) = \\ = \frac{ { \tg }^{7} (x)}{7} + \frac{ { \tg }^{5}(x) }{5} + C$