What is the integral of (sin(x)+cos(x))/(tan(x)) ?

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

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integral of (sin(x)+cos(x))/(tan(x))

\int \frac{sin ( x ) + cos ( x )}{tan ( x )} d x

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

Solution steps

\int \frac{sin ( x ) + cos ( x )}{tan ( x )} d x

Expand

\frac{sin ( x ) + cos ( x )}{tan ( x )} : \frac{sin ( x )}{tan ( x )} + \frac{cos ( x )}{tan ( x )}

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

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

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

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

Add a constant to the solution

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

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\frac{\partial}{\partial x} (\frac{10 x}{1 + x ^{2} - y})

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