What is the integral of sec(5t)(sec(5t)+tan(5t)) ?

The integral of sec(5t)(sec(5t)+tan(5t)) is 1/5 tan(5t)+1/5 sec(5t)+C

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integral of sec(5t)(sec(5t)+tan(5t))

\int sec (5 t) (sec (5 t) + tan (5 t)) d t

\frac{1}{5} tan (5 t) + \frac{1}{5} sec (5 t) + C

Solution steps

\int sec (5 t) (sec (5 t) + tan (5 t)) d t

Expand

sec (5 t) (sec (5 t) + tan (5 t)) : sec^{2} (5 t) + sec (5 t) tan (5 t)

= \int sec^{2} (5 t) + sec (5 t) tan (5 t) d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int sec^{2} (5 t) d t + \int sec (5 t) tan (5 t) d t

\int sec^{2} (5 t) d t = \frac{1}{5} tan (5 t)

\int sec (5 t) tan (5 t) d t = \frac{1}{5} sec (5 t)

= \frac{1}{5} tan (5 t) + \frac{1}{5} sec (5 t)

Add a constant to the solution

= \frac{1}{5} tan (5 t) + \frac{1}{5} sec (5 t) + C

What is the integral of sec(5t)(sec(5t)+tan(5t)) ?
The integral of sec(5t)(sec(5t)+tan(5t)) is 1/5 tan(5t)+1/5 sec(5t)+C