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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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} ( x )} d x + \int \frac{5 tan ( x )}{cos ( x )} d x

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

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

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

Add a constant to the solution

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

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