What is the integral of 3(cos(x)+sin(2x))/(sin(x)) ?

The integral of 3(cos(x)+sin(2x))/(sin(x)) is 3(ln|sin(x)|+2sin(x))+C

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integral of 3(cos(x)+sin(2x))/(sin(x))

\int 3 \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

3 (ln ∣ sin (x) ∣ + 2 sin (x)) + C

Solution steps

\int 3 \cdot \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

Expand

\frac{cos ( x ) + sin ( 2 x )}{sin ( x )} : \frac{cos ( x )}{sin ( x )} + \frac{sin ( 2 x )}{sin ( x )}

= 3 \cdot \int \frac{cos ( x )}{sin ( x )} + \frac{sin ( 2 x )}{sin ( 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

= 3 (\int \frac{cos ( x )}{sin ( x )} d x + \int \frac{sin ( 2 x )}{sin ( x )} d x)

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

\int \frac{sin ( 2 x )}{sin ( x )} d x = 2 sin (x)

= 3 (ln ∣ sin (x) ∣ + 2 sin (x))

Add a constant to the solution

= 3 (ln ∣ sin (x) ∣ + 2 sin (x)) + C

x \to 0 lim ((1 + x)^{x})

\frac{\partial}{\partial x} (x y e^{3 y})

\frac{d}{d x} (10 x^{\frac{1}{5}})

\frac{d}{d x} (\frac{x ^{2}}{x + 2})

x^{\frac{11}{2}}, x = 1

What is the integral of 3(cos(x)+sin(2x))/(sin(x)) ?
The integral of 3(cos(x)+sin(2x))/(sin(x)) is 3(ln|sin(x)|+2sin(x))+C