What is the integral of (7sin(x)+4sec(x))/(tan(x)) ?

The integral of (7sin(x)+4sec(x))/(tan(x)) is 7sin(x)+4ln|tan(x/2)|+C

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integral of (7sin(x)+4sec(x))/(tan(x))

\int \frac{7 sin ( x ) + 4 sec ( x )}{tan ( x )} d x

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

Solution steps

\int \frac{7 sin ( x ) + 4 sec ( x )}{tan ( x )} d x

Expand

\frac{7 sin ( x ) + 4 sec ( x )}{tan ( x )} : \frac{7 sin ( x )}{tan ( x )} + \frac{4 sec ( x )}{tan ( x )}

= \int \frac{7 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{7 sin ( x )}{tan ( x )} d x + \int \frac{4 sec ( x )}{tan ( x )} d x

\int \frac{7 sin ( x )}{tan ( x )} d x = 7 sin (x)

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

= 7 sin (x) + 4 ln tan (\frac{x}{2})

Add a constant to the solution

= 7 sin (x) + 4 ln tan (\frac{x}{2}) + C

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What is the integral of (7sin(x)+4sec(x))/(tan(x)) ?
The integral of (7sin(x)+4sec(x))/(tan(x)) is 7sin(x)+4ln|tan(x/2)|+C