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

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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_{0}^{\frac{1}{2}} 7 arccos (x) d x

y^{^{'}} = 14 x y - 2 x

\frac{\partial}{\partial y} (\frac{1 + y}{1 + x})

\frac{\partial}{\partial x} (\frac{a}{sin ( x )})

y^{^{'}} + y cos (x) = 5 cos (x), y (0) = 7

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