What is the integral of (4xsin(x)-5csc(x))/(sin(x)) ?

The integral of (4xsin(x)-5csc(x))/(sin(x)) is 2x^2+5cot(x)+C

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integral of (4xsin(x)-5csc(x))/(sin(x))

\int \frac{4 x sin ( x ) - 5 csc ( x )}{sin ( x )} d x

2 x^{2} + 5 cot (x) + C

Solution steps

\int \frac{4 x sin ( x ) - 5 csc ( x )}{sin ( x )} d x

Expand

\frac{4 x sin ( x ) - 5 csc ( x )}{sin ( x )} : 4 x - \frac{5 csc ( x )}{sin ( x )}

= \int 4 x - \frac{5 csc ( 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

= \int 4 x d x - \int \frac{5 csc ( x )}{sin ( x )} d x

\int 4 x d x = 2 x^{2}

\int \frac{5 csc ( x )}{sin ( x )} d x = - 5 cot (x)

= 2 x^{2} - (- 5 cot (x))

Simplify

= 2 x^{2} + 5 cot (x)

Add a constant to the solution

= 2 x^{2} + 5 cot (x) + C

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What is the integral of (4xsin(x)-5csc(x))/(sin(x)) ?
The integral of (4xsin(x)-5csc(x))/(sin(x)) is 2x^2+5cot(x)+C