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Beliebt Rechnen >

integral of (cos(x+pi/2))/(5sin(x+pi/2))

Lösung

\int \frac{cos ( x + \frac{π}{2} )}{5 sin ( x + \frac{π}{2} )} d x

Lösung

\frac{1}{5} (x cot (x + \frac{π}{2}) + x tan (x) + ln ∣ cos (x) ∣) + C

Schritte zur Lösung

\int \frac{cos ( x + \frac{π}{2} )}{5 sin ( x + \frac{π}{2} )} d x

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int \frac{cos ( x + \frac{π}{2} )}{sin ( x + \frac{π}{2} )} d x

Wende die partielle Integration an

= \frac{1}{5} (x cot (x + \frac{π}{2}) - \int - x sec^{2} (x) d x)

\int - x sec^{2} (x) d x = - x tan (x) - ln ∣ cos (x) ∣

= \frac{1}{5} (x cot (x + \frac{π}{2}) - (- x tan (x) - ln ∣ cos (x) ∣))

Vereinfache

= \frac{1}{5} (x cot (x + \frac{π}{2}) + x tan (x) + ln ∣ cos (x) ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (x cot (x + \frac{π}{2}) + x tan (x) + ln ∣ cos (x) ∣) + C

Graph

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