What is the integral of sec(x)(2tan(x)-6sec(x)) ?

The integral of sec(x)(2tan(x)-6sec(x)) is 2sec(x)-6tan(x)+C

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integral of sec(x)(2tan(x)-6sec(x))

\int sec (x) (2 tan (x) - 6 sec (x)) d x

2 sec (x) - 6 tan (x) + C

Solution steps

\int sec (x) (2 tan (x) - 6 sec (x)) d x

Rewrite using trig identities

= \int \frac{2 sin ( x ) - 6}{cos ^{2} ( x )} d x

Expand

\frac{2 sin ( x ) - 6}{cos ^{2} ( x )} : \frac{2 sin ( x )}{cos ^{2} ( x )} - \frac{6}{cos ^{2} ( x )}

= \int \frac{2 sin ( x )}{cos ^{2} ( x )} - \frac{6}{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{2 sin ( x )}{cos ^{2} ( x )} d x - \int \frac{6}{cos ^{2} ( x )} d x

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

\int \frac{6}{cos ^{2} ( x )} d x = 6 tan (x)

= 2 sec (x) - 6 tan (x)

Add a constant to the solution

= 2 sec (x) - 6 tan (x) + C

\int \frac{x ^{2} + 24 x - 4}{x ^{3} - 4 x} d x

\frac{\partial}{\partial x} (\frac{n}{x + 1})

x \to - 5 lim (\frac{5}{x + 3})

\frac{\partial}{\partial x} (- 1)

\int \frac{3 x ^{11}}{x ^{2} - x - 6} d x

What is the integral of sec(x)(2tan(x)-6sec(x)) ?
The integral of sec(x)(2tan(x)-6sec(x)) is 2sec(x)-6tan(x)+C