What is the integral of sec(t)(2sec(t)+5tan(t)) ?

The integral of sec(t)(2sec(t)+5tan(t)) is 2tan(t)+5sec(t)+C

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integral of sec(t)(2sec(t)+5tan(t))

\int sec (t) (2 sec (t) + 5 tan (t)) d t

2 tan (t) + 5 sec (t) + C

Solution steps

\int sec (t) (2 sec (t) + 5 tan (t)) d t

Rewrite using trig identities

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

Expand

\frac{2 + 5 tan ( t ) cos ( t )}{cos ^{2} ( t )} : \frac{2}{cos ^{2} ( t )} + \frac{5 tan ( t )}{cos ( t )}

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

\int \frac{2}{cos ^{2} ( t )} d t = 2 tan (t)

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

= 2 tan (t) + 5 sec (t)

Add a constant to the solution

= 2 tan (t) + 5 sec (t) + C

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What is the integral of sec(t)(2sec(t)+5tan(t)) ?
The integral of sec(t)(2sec(t)+5tan(t)) is 2tan(t)+5sec(t)+C