integral of cos^2(xsi)n^3x

Lösung

\int cos^{2} (x s i) n^{3} x d x

\frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x)) + C

Schritte zur Lösung

\int cos^{2} (x s i) n^{3} x d x

Entferne die Konstante:

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

= n^{3} \cdot \int cos^{2} (s i x) x d x

Wende U-Substitution an

= n^{3} \cdot \int \frac{u cos ^{2} ( u )}{s ^{2} i ^{2}} d u

Entferne die Konstante:

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

= n^{3} \frac{1}{s ^{2} i ^{2}} \cdot \int u cos^{2} (u) d u

Wende die partielle Integration an

= n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} u (u + \frac{1}{2} sin (2 u)) - \int \frac{1}{2} (u + \frac{1}{2} sin (2 u)) d u)

\int \frac{1}{2} (u + \frac{1}{2} sin (2 u)) d u = \frac{1}{2} (\frac{u ^{2}}{2} - \frac{1}{4} cos (2 u))

= n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} u (u + \frac{1}{2} sin (2 u)) - \frac{1}{2} (\frac{u ^{2}}{2} - \frac{1}{4} cos (2 u)))

Setze in

u = s i x

ein

= n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} s i x (s i x + \frac{1}{2} sin (2 s i x)) - \frac{1}{2} (\frac{( s i x ) ^{2}}{2} - \frac{1}{4} cos (2 s i x)))

Vereinfache

n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} s i x (s i x + \frac{1}{2} sin (2 s i x)) - \frac{1}{2} (\frac{( s i x ) ^{2}}{2} - \frac{1}{4} cos (2 s i x))) : \frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x))

= \frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x))

Füge eine Konstante zur Lösung hinzu

= \frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x)) + C

t an g e n t e^{- 2 t}, at t = 0

\frac{d}{d x} (\frac{x}{x - 2 x})

x \to - 1 lim (2 x^{2} + 6 x - 1)

x \to 2 lim (\frac{5}{∣ 2 - x ∣})

\frac{d}{d x} (e^{- \frac{x}{a}})