What is the integral of sec(x/3)-csc(5x) ?

The integral of sec(x/3)-csc(5x) is 3ln|tan(x/3)+sec(x/3)|-1/5 ln|tan((5x)/2)|+C

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integral of sec(x/3)-csc(5x)

\int sec (\frac{x}{3}) - csc (5 x) d x

3 ln tan (\frac{x}{3}) + sec (\frac{x}{3}) - \frac{1}{5} ln tan (\frac{5 x}{2}) + C

Solution steps

\int sec (\frac{x}{3}) - 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 sec (\frac{x}{3}) d x - \int csc (5 x) d x

\int sec (\frac{x}{3}) d x = 3 ln tan (\frac{x}{3}) + sec (\frac{x}{3})

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

= 3 ln tan (\frac{x}{3}) + sec (\frac{x}{3}) - \frac{1}{5} ln tan (\frac{5 x}{2})

Add a constant to the solution

= 3 ln tan (\frac{x}{3}) + sec (\frac{x}{3}) - \frac{1}{5} ln tan (\frac{5 x}{2}) + C

\frac{d}{d x} (sec^{5} (x^{7} + 4 x)^{6})

x^{2020} + \frac{1}{3} x^{3} - 4 x^{2}

\frac{\partial}{\partial y} (x e^{2 x + 3 y})

\int (\frac{cos ( x )}{sin ^{2} ( x )}) d x

\int u d u

What is the integral of sec(x/3)-csc(5x) ?
The integral of sec(x/3)-csc(5x) is 3ln|tan(x/3)+sec(x/3)|-1/5 ln|tan((5x)/2)|+C