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

解

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

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

解答ステップ

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

三角関数の公式を使用して書き換える

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

総和規則を適用する:

\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) ∣

定数を解答に追加する

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