integral of cot^2(xcs)c^4x

Lösung

\int cot^{2} (x cs) c^{4} x d x

\frac{c ^{2}}{2 s ^{2}} (- c^{2} s^{2} x^{2} - 2 cs x cot (cs x) + 2 ln ∣ sin (cs x) ∣) + C

Schritte zur Lösung

\int cot^{2} (x cs) c^{4} x d x

Entferne die Konstante:

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

= c^{4} \cdot \int cot^{2} (cs x) x d x

Wende U-Substitution an

= c^{4} \cdot \int \frac{u cot ^{2} ( u )}{c ^{2} s ^{2}} d u

Entferne die Konstante:

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

= c^{4} \frac{1}{c ^{2} s ^{2}} \cdot \int u cot^{2} (u) d u

Wende die partielle Integration an

= c^{4} \frac{1}{c ^{2} s ^{2}} (u (- u - cot (u)) - \int - u - cot (u) d u)

\int - u - cot (u) d u = - \frac{u ^{2}}{2} - ln ∣ sin (u) ∣

= c^{4} \frac{1}{c ^{2} s ^{2}} (u (- u - cot (u)) - (- \frac{u ^{2}}{2} - ln ∣ sin (u) ∣))

Setze in

u = cs x

ein

= c^{4} \frac{1}{c ^{2} s ^{2}} (cs x (- cs x - cot (cs x)) - (- \frac{( cs x ) ^{2}}{2} - ln ∣ sin (cs x) ∣))

Vereinfache

c^{4} \frac{1}{c ^{2} s ^{2}} (cs x (- cs x - cot (cs x)) - (- \frac{( cs x ) ^{2}}{2} - ln ∣ sin (cs x) ∣)) : \frac{c ^{2}}{2 s ^{2}} (- c^{2} s^{2} x^{2} - 2 cs x cot (cs x) + 2 ln ∣ sin (cs x) ∣)

= \frac{c ^{2}}{2 s ^{2}} (- c^{2} s^{2} x^{2} - 2 cs x cot (cs x) + 2 ln ∣ sin (cs x) ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{c ^{2}}{2 s ^{2}} (- c^{2} s^{2} x^{2} - 2 cs x cot (cs x) + 2 ln ∣ sin (cs x) ∣) + C

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

\frac{d}{d x} (\frac{cos ( y ) + 2 e ^{- x} cos ( x )}{y})

t an g e n t y = 3 x e^{x}, (0, 0)

y^{^{'}} = x y + x

\frac{d y}{d t} + 6 y = 0