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

\int_{1}^{9} f (x) d x

x \to - 1 lim (arctan (\frac{x - 1}{x + 1}))

6 x^{3}

(5 - 7 x)^{3}

12 (x - \frac{1}{x})^{2}, at x = 2

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