What is the integral of sec(x)(8tan(x)+2sec(x)) ?

The integral of sec(x)(8tan(x)+2sec(x)) is 8sec(x)+2tan(x)+C

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integral of sec(x)(8tan(x)+2sec(x))

\int sec (x) (8 tan (x) + 2 sec (x)) d x

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

Solution steps

\int sec (x) (8 tan (x) + 2 sec (x)) d x

Rewrite using trig identities

= \int \frac{8 tan ( x ) cos ( x ) + 2}{cos ^{2} ( x )} d x

Expand

\frac{8 tan ( x ) cos ( x ) + 2}{cos ^{2} ( x )} : \frac{8 tan ( x )}{cos ( x )} + \frac{2}{cos ^{2} ( x )}

= \int \frac{8 tan ( x )}{cos ( x )} + \frac{2}{cos ^{2} ( 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{8 tan ( x )}{cos ( x )} d x + \int \frac{2}{cos ^{2} ( x )} d x

\int \frac{8 tan ( x )}{cos ( x )} d x = 8 sec (x)

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

= 8 sec (x) + 2 tan (x)

Add a constant to the solution

= 8 sec (x) + 2 tan (x) + C

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What is the integral of sec(x)(8tan(x)+2sec(x)) ?
The integral of sec(x)(8tan(x)+2sec(x)) is 8sec(x)+2tan(x)+C