Помогите найти ошибку

[tex] \int\limits6^{sin(x)} dx \\\\t = sin(x)\\\\dt = d(sin(x) ) = (sin(x))`dx = cos(x)dx\\\\dt = cos(x)dx\\\\dx=\frac{dt}{cos(x)} \\\\\int\limits6^t*\frac{1}{cos(x)}*dt = \frac{1}{cos(x)}\int\limits6^tdt = sec(x)\int\limits6^tdt= sec(x)*\frac{6^t}{In(6)} = \frac{sec(x)6^{sin(x)}}{In(6)}[/tex]

При дифференцировании полученной первообразной я получил абсолютно другую функцию:

[tex](\frac{sec(x)6^{sin(x)}}{In(6)})` = \frac{1}{In(6)}(sec(x)6^{sin(x)})` = \frac{1}{In(6)}[(sec(x))`6^{sin(x)} + (6^{sin(x)})`sec(x)] =\\= \frac{1}{In(6)}[tg(x)sec(x)6^{sin(x)} + In(6)cos(x)6^{sin(x)}sec(x)] = \\\\ = \frac{1}{In(6)}[\frac{6^{sin(x)}sin(x) }{cos^2(x)} + In(6)6^{sin(x)}] = \frac{6^{sin(x)}sin(x) + In(6)6^{sin(x)}cos^2(x) }{In(6)cos^2(x)} = \\\\= \frac{6^{sin(x)}sin(x)}{In(6)cos^2(x)} + 6^{sin(x)}[/tex]

Вопрос: где я допустил ошибку во время интегрирования?
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