What is the integral of (1-(x+2)cot(x))/(sin(x)) ?

The integral of (1-(x+2)cot(x))/(sin(x)) is xcsc(x)+2csc(x)+C

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integral of (1-(x+2)cot(x))/(sin(x))

\int \frac{1 - ( x + 2 ) cot ( x )}{sin ( x )} d x

x csc (x) + 2 csc (x) + C

Solution steps

\int \frac{1 - ( x + 2 ) cot ( x )}{sin ( x )} d x

Expand

\frac{1 - ( x + 2 ) cot ( x )}{sin ( x )} : \frac{1}{sin ( x )} - \frac{x cot ( x )}{sin ( x )} - \frac{2 cot ( x )}{sin ( x )}

= \int \frac{1}{sin ( x )} - \frac{x cot ( x )}{sin ( x )} - \frac{2 cot ( 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 \frac{1}{sin ( x )} d x - \int \frac{x cot ( x )}{sin ( x )} d x - \int \frac{2 cot ( x )}{sin ( x )} d x

\int \frac{1}{sin ( x )} d x = ln tan (\frac{x}{2})

\int \frac{x cot ( x )}{sin ( x )} d x = - x csc (x) + ln tan (\frac{x}{2})

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

= ln tan (\frac{x}{2}) - (- x csc (x) + ln tan (\frac{x}{2})) - (- 2 csc (x))

Simplify

ln tan (\frac{x}{2}) - (- x csc (x) + ln tan (\frac{x}{2})) - (- 2 csc (x)) : x csc (x) + 2 csc (x)

= x csc (x) + 2 csc (x)

Add a constant to the solution

= x csc (x) + 2 csc (x) + C

d er i v a t i v e lo g_{6} (x^{2} - 5 x)

\int_{0}^{\frac{π}{3}} 7 sec^{2} (x) d x

\frac{\partial}{\partial x} (3 ln (x))

a re a (x + 6), x, - 2, 3

\int sec^{6} (x) tan (x) d x

What is the integral of (1-(x+2)cot(x))/(sin(x)) ?
The integral of (1-(x+2)cot(x))/(sin(x)) is xcsc(x)+2csc(x)+C