What is the integral of 5(cos(x)+sin(2x))/(sin(x)) ?

The integral of 5(cos(x)+sin(2x))/(sin(x)) is 5(ln|sin(x)|+2sin(x))+C

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integral of 5(cos(x)+sin(2x))/(sin(x))

\int 5 \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

5 (ln ∣ sin (x) ∣ + 2 sin (x)) + C

Solution steps

\int 5 \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

= 5 \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 )}

= 5 \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

= 5 (\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)

= 5 (ln ∣ sin (x) ∣ + 2 sin (x))

Add a constant to the solution

= 5 (ln ∣ sin (x) ∣ + 2 sin (x)) + C

n = 1 \sum \infty 8 (- \frac{1}{4})^{n}

x \to 0 + lim (2 x ln (x))

x^{2} + 4, - x^{2} - 1, x = - 2, x = 3

f (x) = x^{2} ln (2 x)

x y^{^{'}} = y + 5 x^{2} sin (x)

What is the integral of 5(cos(x)+sin(2x))/(sin(x)) ?
The integral of 5(cos(x)+sin(2x))/(sin(x)) is 5(ln|sin(x)|+2sin(x))+C