integral of tan^2(xco)s^3x

Lösung

\int tan^{2} (x co) s^{3} x d x

\frac{s ^{3}}{2 c ^{2} o ^{2}} (- c^{2} o^{2} x^{2} + 2 co x tan (co x) + 2 ln ∣ cos (co x) ∣) + C

Schritte zur Lösung

\int tan^{2} (x co) s^{3} x d x

Entferne die Konstante:

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

= s^{3} \cdot \int tan^{2} (co x) x d x

Wende U-Substitution an

= s^{3} \cdot \int \frac{u tan ^{2} ( u )}{c ^{2} o ^{2}} d u

Entferne die Konstante:

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

= s^{3} \frac{1}{c ^{2} o ^{2}} \cdot \int u tan^{2} (u) d u

Wende die partielle Integration an

= s^{3} \frac{1}{c ^{2} o ^{2}} (u (- u + tan (u)) - \int - u + tan (u) d u)

\int - u + tan (u) d u = - \frac{u ^{2}}{2} - ln ∣ cos (u) ∣

= s^{3} \frac{1}{c ^{2} o ^{2}} (u (- u + tan (u)) - (- \frac{u ^{2}}{2} - ln ∣ cos (u) ∣))

Setze in

u = co x

ein

= s^{3} \frac{1}{c ^{2} o ^{2}} (co x (- co x + tan (co x)) - (- \frac{( co x ) ^{2}}{2} - ln ∣ cos (co x) ∣))

Vereinfache

s^{3} \frac{1}{c ^{2} o ^{2}} (co x (- co x + tan (co x)) - (- \frac{( co x ) ^{2}}{2} - ln ∣ cos (co x) ∣)) : \frac{s ^{3}}{2 c ^{2} o ^{2}} (- c^{2} o^{2} x^{2} + 2 co x tan (co x) + 2 ln ∣ cos (co x) ∣)

= \frac{s ^{3}}{2 c ^{2} o ^{2}} (- c^{2} o^{2} x^{2} + 2 co x tan (co x) + 2 ln ∣ cos (co x) ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{s ^{3}}{2 c ^{2} o ^{2}} (- c^{2} o^{2} x^{2} + 2 co x tan (co x) + 2 ln ∣ cos (co x) ∣) + C

x \to - 2 lim (\frac{- 3}{x + 2})

\int e^{x + 3 e^{x}} d x

\frac{d ^{2} y}{d x ^{2}} - 9 y = 0

\frac{\partial}{\partial x} (a - b x - \frac{c}{y} + d \frac{x}{y})

x \to 2 lim (2 x^{2} - x + 2)