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

The integral of (pisec(x)-tan(x))sec(x) is pitan(x)-sec(x)+C

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integral of (pisec(x)-tan(x))sec(x)

\int (π sec (x) - tan (x)) sec (x) d x

π tan (x) - sec (x) + C

Solution steps

\int (π sec (x) - tan (x)) sec (x) d x

Rewrite using trig identities

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

Expand

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

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

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

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

= π tan (x) - sec (x)

Add a constant to the solution

= π tan (x) - sec (x) + C

What is the integral of (pisec(x)-tan(x))sec(x) ?
The integral of (pisec(x)-tan(x))sec(x) is pitan(x)-sec(x)+C