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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

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 t + \int \frac{cos ( t ) - sin ( t )}{2 ( sin ( t ) + cos ( t ) )} d t

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

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

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

Add a constant to the solution

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

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