What is the integral of (tan(x)-sin(x)sin(y)) ?

The integral of (tan(x)-sin(x)sin(y)) is -ln|cos(x)|+sin(y)cos(x)+C

integral of (tan(x)-sin(x)sin(y))

\int (tan (x) - sin (x) sin (y)) d x

- ln ∣ cos (x) ∣ + sin (y) cos (x) + C

Solution steps

\int tan (x) - sin (x) sin (y) 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 tan (x) d x - \int sin (y) sin (x) d x

\int tan (x) d x = - ln ∣ cos (x) ∣

\int sin (y) sin (x) d x = - sin (y) cos (x)

= - ln ∣ cos (x) ∣ - (- sin (y) cos (x))

Simplify

= - ln ∣ cos (x) ∣ + sin (y) cos (x)

Add a constant to the solution

= - ln ∣ cos (x) ∣ + sin (y) cos (x) + C

\int \frac{x ^{3} + 1}{x ^{2}} d x

\frac{d}{d x} (\frac{3}{x} + 5)

\int (\frac{1}{x ln ( x )}) d x

\frac{\partial}{\partial x} (x y^{2} - 5 y + 6)

f (x) = 6 x sin (x), at x = \frac{π}{2}

What is the integral of (tan(x)-sin(x)sin(y)) ?
The integral of (tan(x)-sin(x)sin(y)) is -ln|cos(x)|+sin(y)cos(x)+C