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

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

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

\int (tan (x) - sec (x))^{2} d x

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

Solution steps

\int (tan (x) - sec (x))^{2} d x

Rewrite using trig identities

= \int \frac{( sin ( x ) - 1 ) ^{2}}{cos ^{2} ( x )} d x

Expand

\frac{( sin ( x ) - 1 ) ^{2}}{cos ^{2} ( x )} : \frac{sin ^{2} ( x )}{cos ^{2} ( x )} - \frac{2 sin ( x )}{cos ^{2} ( x )} + \frac{1}{cos ^{2} ( x )}

= \int \frac{sin ^{2} ( x )}{cos ^{2} ( x )} - \frac{2 sin ( x )}{cos ^{2} ( x )} + \frac{1}{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{sin ^{2} ( x )}{cos ^{2} ( x )} d x - \int \frac{2 sin ( x )}{cos ^{2} ( x )} d x + \int \frac{1}{cos ^{2} ( x )} d x

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

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

\int \frac{1}{cos ^{2} ( x )} d x = tan (x)

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

Simplify

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

Add a constant to the solution

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

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