What is the integral of [12+csc(x)(sin(x)+csc(x))] ?

The integral of [12+csc(x)(sin(x)+csc(x))] is -cot(x)+13x+C

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integral of [12+csc(x)(sin(x)+csc(x))]

\int [12 + csc (x) (sin (x) + csc (x))] d x

- cot (x) + 13 x + C

Solution steps

\int 12 + csc (x) (sin (x) + csc (x)) d x

Expand

csc (x) (sin (x) + csc (x)) : csc (x) sin (x) + csc^{2} (x)

= \int 12 + csc (x) sin (x) + csc^{2} (x) d x

Simplify

12 + csc (x) sin (x) + csc^{2} (x) : csc^{2} (x) + 13

= \int csc^{2} (x) + 13 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 csc^{2} (x) d x + \int 13 d x

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

\int 13 d x = 13 x

= - cot (x) + 13 x

Add a constant to the solution

= - cot (x) + 13 x + C

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

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\int \frac{t ^{2}}{t ^{2} + 4} d t

\int \frac{1}{3} x^{4} d x

\int \frac{1}{cos ( 2 t )} d t

What is the integral of [12+csc(x)(sin(x)+csc(x))] ?
The integral of [12+csc(x)(sin(x)+csc(x))] is -cot(x)+13x+C