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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

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

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

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

Add a constant to the solution

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

\frac{d}{d x} ((1 + 3 x)^{\frac{1}{2 x}})

\frac{( 3 y ^{2} - t ^{2} )}{y ^{5}} (\frac{d y}{d t}) + \frac{t}{2 y ^{4}} = 0

x, at x = 9

\frac{\partial}{\partial y} (4 y)

(10, 5), (1, 12)

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