What is the integral of sec(y)(tan(y)-sec(y)) ?

The integral of sec(y)(tan(y)-sec(y)) is sec(y)-tan(y)+C

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integral of sec(y)(tan(y)-sec(y))

\int sec (y) (tan (y) - sec (y)) d y

sec (y) - tan (y) + C

Solution steps

\int sec (y) (tan (y) - sec (y)) d y

Rewrite using trig identities

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

Expand

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

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

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 ( y )}{cos ^{2} ( y )} d y - \int \frac{1}{cos ^{2} ( y )} d y

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

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

= sec (y) - tan (y)

Add a constant to the solution

= sec (y) - tan (y) + C

\frac{d}{d x} (e^{(x + 1)^{2}})

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f (x) = x^{2} + 3, at x = \frac{1}{2}

\frac{\partial}{\partial y} (cos (x z))

\int (3 t e^{3 t}) d t

What is the integral of sec(y)(tan(y)-sec(y)) ?
The integral of sec(y)(tan(y)-sec(y)) is sec(y)-tan(y)+C