What is the integral of (5sin(x)+2sec(x))/(tan(x)) ?

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

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integral of (5sin(x)+2sec(x))/(tan(x))

\int \frac{5 sin ( x ) + 2 sec ( x )}{tan ( x )} d x

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

Solution steps

\int \frac{5 sin ( x ) + 2 sec ( x )}{tan ( x )} d x

Expand

\frac{5 sin ( x ) + 2 sec ( x )}{tan ( x )} : \frac{5 sin ( x )}{tan ( x )} + \frac{2 sec ( x )}{tan ( x )}

= \int \frac{5 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{5 sin ( x )}{tan ( x )} d x + \int \frac{2 sec ( x )}{tan ( x )} d x

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

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

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

Add a constant to the solution

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

e^{3 t} sin (2 t)

\int_{0}^{0.6} 600 x d x

\int \frac{1}{500} e^{- \frac{1}{500} t} d t

\frac{\partial}{\partial y} (8 x + 6 x^{2} y + 2 y^{2})

\frac{1}{8}

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