integral of sqrt(cos^2(x)+cos(x))

Lösung

\int cos^{2} (x) + cos (x) d x

- i (\frac{cos ( x ) - 1 + 1}{cos ( x ) - 1} (cos (x) - 1) - (- ln cos (x) - 1 + 1 + cos (x) - 1)) + C

Schritte zur Lösung

\int cos^{2} (x) + cos (x) d x

Wende U-Substitution an

= \int - i \frac{u}{u - 1} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - i \cdot \int \frac{u}{u - 1} d u

Wende U-Substitution an

= - i \cdot \int \frac{v + 1}{v} d v

Wende die partielle Integration an

= - i (\frac{v + 1}{v} v - \int - \frac{1}{2 v ^{\frac{1}{2}} v + 1} d v)

\int - \frac{1}{2 v ^{\frac{1}{2}} v + 1} d v = - ln v + 1 + v

= - i (\frac{v + 1}{v} v - (- ln v + 1 + v))

Ersetze zurück

= - i (\frac{cos ( x ) - 1 + 1}{cos ( x ) - 1} (cos (x) - 1) - (- ln cos (x) - 1 + 1 + cos (x) - 1))

Füge eine Konstante zur Lösung hinzu

= - i (\frac{cos ( x ) - 1 + 1}{cos ( x ) - 1} (cos (x) - 1) - (- ln cos (x) - 1 + 1 + cos (x) - 1)) + C