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

The integral of (cos(x))/((cos(x)+sin(x))) is 1/2 x+1/2 ln|cos(x)+sin(x)|+C

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

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

\frac{1}{2} x + \frac{1}{2} ln ∣ cos (x) + sin (x) ∣ + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1}{2} + \frac{cos ( x ) - sin ( x )}{2 ( cos ( x ) + sin ( 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{1}{2} d x + \int \frac{cos ( x ) - sin ( x )}{2 ( cos ( x ) + sin ( x ) )} d x

\int \frac{1}{2} d x = \frac{1}{2} x

\int \frac{cos ( x ) - sin ( x )}{2 ( cos ( x ) + sin ( x ) )} d x = \frac{1}{2} ln ∣ cos (x) + sin (x) ∣

= \frac{1}{2} x + \frac{1}{2} ln ∣ cos (x) + sin (x) ∣

Add a constant to the solution

= \frac{1}{2} x + \frac{1}{2} ln ∣ cos (x) + sin (x) ∣ + C

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What is the integral of (cos(x))/((cos(x)+sin(x))) ?
The integral of (cos(x))/((cos(x)+sin(x))) is 1/2 x+1/2 ln|cos(x)+sin(x)|+C