What is the integral of (5sin(x)+4sec(x))/(tan(x)) ?

The integral of (5sin(x)+4sec(x))/(tan(x)) is 5sin(x)+4ln|tan(x/2)|+C

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integral of (5sin(x)+4sec(x))/(tan(x))

\int \frac{5 sin ( x ) + 4 sec ( x )}{tan ( x )} d x

5 sin (x) + 4 ln tan (\frac{x}{2}) + C

Solution steps

\int \frac{5 sin ( x ) + 4 sec ( x )}{tan ( x )} d x

Expand

\frac{5 sin ( x ) + 4 sec ( x )}{tan ( x )} : \frac{5 sin ( x )}{tan ( x )} + \frac{4 sec ( x )}{tan ( x )}

= \int \frac{5 sin ( x )}{tan ( x )} + \frac{4 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{4 sec ( x )}{tan ( x )} d x

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

\int \frac{4 sec ( x )}{tan ( x )} d x = 4 ln tan (\frac{x}{2})

= 5 sin (x) + 4 ln tan (\frac{x}{2})

Add a constant to the solution

= 5 sin (x) + 4 ln tan (\frac{x}{2}) + C

\frac{d}{d x} (e^{x} (x^{2} + 5 x))

\frac{d}{d x} (x 3 x - 2)

x y^{^{'}} + y = 0, y (4) = 6

f (x) = \frac{4}{x} - \frac{2}{x ^{3}} + \frac{1}{x ^{4}}

y = - \frac{7}{2} x^{2} + 5 x + 5, (- 1, - \frac{7}{2})

What is the integral of (5sin(x)+4sec(x))/(tan(x)) ?
The integral of (5sin(x)+4sec(x))/(tan(x)) is 5sin(x)+4ln|tan(x/2)|+C