What is the integral of (sin(x)-cos(y)+y/x) ?

The integral of (sin(x)-cos(y)+y/x) is -sin(y)+(y^2)/(2x)+sin(x)y+C

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integral of (sin(x)-cos(y)+y/x)

\int (sin (x) - cos (y) + \frac{y}{x}) d y

- sin (y) + \frac{y ^{2}}{2 x} + sin (x) y + C

Solution steps

\int sin (x) - cos (y) + \frac{y}{x} 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 cos (y) d y + \int \frac{y}{x} d y + \int sin (x) d y

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

\int \frac{y}{x} d y = \frac{y ^{2}}{2 x}

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

= - sin (y) + \frac{y ^{2}}{2 x} + sin (x) y

Add a constant to the solution

= - sin (y) + \frac{y ^{2}}{2 x} + sin (x) y + C

\int (\frac{a ^{x} + b ^{x}}{c ^{x}}) d x

\int x^{3} x^{2} - 100 d x

\int x^{3} 1 + 81 x^{2} d x

\int (\frac{x - 1}{x})^{3} d x

\int x tan (1 - x^{2}) d x

What is the integral of (sin(x)-cos(y)+y/x) ?
The integral of (sin(x)-cos(y)+y/x) is -sin(y)+(y^2)/(2x)+sin(x)y+C