What is the integral of (cos(ax))/(sin(ax))+sin(bx) ?

The integral of (cos(ax))/(sin(ax))+sin(bx) is 1/a ln|sin(ax)|-1/b cos(bx)+C

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integral of (cos(ax))/(sin(ax))+sin(bx)

\int \frac{cos ( a x )}{sin ( a x )} + sin (b x) d x

\frac{1}{a} ln ∣ sin (a x) ∣ - \frac{1}{b} cos (b x) + C

Solution steps

\int \frac{cos ( a x )}{sin ( a x )} + sin (b x) 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 \frac{cos ( a x )}{sin ( a x )} d x + \int sin (b x) d x

\int \frac{cos ( a x )}{sin ( a x )} d x = \frac{1}{a} ln ∣ sin (a x) ∣

\int sin (b x) d x = - \frac{1}{b} cos (b x)

= \frac{1}{a} ln ∣ sin (a x) ∣ - \frac{1}{b} cos (b x)

Add a constant to the solution

= \frac{1}{a} ln ∣ sin (a x) ∣ - \frac{1}{b} cos (b x) + C

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What is the integral of (cos(ax))/(sin(ax))+sin(bx) ?
The integral of (cos(ax))/(sin(ax))+sin(bx) is 1/a ln|sin(ax)|-1/b cos(bx)+C