What is the integral of csc(4x)cot(4x)+(13)/(x^2+16) ?

The integral of csc(4x)cot(4x)+(13)/(x^2+16) is -1/4 csc(4x)+13/4 arctan(x/4)+C

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integral of csc(4x)cot(4x)+(13)/(x^2+16)

\int csc (4 x) cot (4 x) + \frac{13}{x ^{2} + 16} d x

- \frac{1}{4} csc (4 x) + \frac{13}{4} arctan (\frac{x}{4}) + C

Solution steps

\int csc (4 x) cot (4 x) + \frac{13}{x ^{2} + 16} 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 (4 x) cot (4 x) d x + \int \frac{13}{x ^{2} + 16} d x

\int csc (4 x) cot (4 x) d x = - \frac{1}{4} csc (4 x)

\int \frac{13}{x ^{2} + 16} d x = \frac{13}{4} arctan (\frac{x}{4})

= - \frac{1}{4} csc (4 x) + \frac{13}{4} arctan (\frac{x}{4})

Add a constant to the solution

= - \frac{1}{4} csc (4 x) + \frac{13}{4} arctan (\frac{x}{4}) + C

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What is the integral of csc(4x)cot(4x)+(13)/(x^2+16) ?
The integral of csc(4x)cot(4x)+(13)/(x^2+16) is -1/4 csc(4x)+13/4 arctan(x/4)+C