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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

= tan (y) - y - sec (y)

Add a constant to the solution

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

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