What is the integral of (cot(5x)+csc(5x)) ?

The integral of (cot(5x)+csc(5x)) is 1/5 ln|sin(5x)|+1/5 ln|tan((5x)/2)|+C

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integral of (cot(5x)+csc(5x))

\int (cot (5 x) + csc (5 x)) d x

\frac{1}{5} ln ∣ sin (5 x) ∣ + \frac{1}{5} ln tan (\frac{5 x}{2}) + C

Solution steps

\int cot (5 x) + csc (5 x) d x

Rewrite using trig identities

= \int \frac{1}{tan ( 5 x )} + csc (5 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}{tan ( 5 x )} d x + \int csc (5 x) d x

\int \frac{1}{tan ( 5 x )} d x = \frac{1}{5} ln ∣ sin (5 x) ∣

\int csc (5 x) d x = \frac{1}{5} ln tan (\frac{5 x}{2})

= \frac{1}{5} ln ∣ sin (5 x) ∣ + \frac{1}{5} ln tan (\frac{5 x}{2})

Add a constant to the solution

= \frac{1}{5} ln ∣ sin (5 x) ∣ + \frac{1}{5} ln tan (\frac{5 x}{2}) + C

What is the integral of (cot(5x)+csc(5x)) ?
The integral of (cot(5x)+csc(5x)) is 1/5 ln|sin(5x)|+1/5 ln|tan((5x)/2)|+C