What is the integral of (3sin(x)+2sec(x))/(tan(x)) ?

The integral of (3sin(x)+2sec(x))/(tan(x)) is 3sin(x)+2ln|tan(x/2)|+C

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integral of (3sin(x)+2sec(x))/(tan(x))

\int \frac{3 sin ( x ) + 2 sec ( x )}{tan ( x )} d x

3 sin (x) + 2 ln tan (\frac{x}{2}) + C

Solution steps

\int \frac{3 sin ( x ) + 2 sec ( x )}{tan ( x )} d x

Expand

\frac{3 sin ( x ) + 2 sec ( x )}{tan ( x )} : \frac{3 sin ( x )}{tan ( x )} + \frac{2 sec ( x )}{tan ( x )}

= \int \frac{3 sin ( x )}{tan ( x )} + \frac{2 sec ( x )}{tan ( 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{3 sin ( x )}{tan ( x )} d x + \int \frac{2 sec ( x )}{tan ( x )} d x

\int \frac{3 sin ( x )}{tan ( x )} d x = 3 sin (x)

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

= 3 sin (x) + 2 ln tan (\frac{x}{2})

Add a constant to the solution

= 3 sin (x) + 2 ln tan (\frac{x}{2}) + C

- \frac{5}{x ^{6}}

ln (2^{x})

\int \frac{t ^{2}}{1 + t ^{2}} d t

\int 8 x^{3} + \frac{1}{x} d x

\frac{\partial}{\partial y} (e^{7 x e^{y}})

What is the integral of (3sin(x)+2sec(x))/(tan(x)) ?
The integral of (3sin(x)+2sec(x))/(tan(x)) is 3sin(x)+2ln|tan(x/2)|+C