integral of cos^2(xsi)n^4x

Lösung

\int cos^{2} (x s i) n^{4} x d x

\frac{n ^{4}}{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^{4} x d x

Entferne die Konstante:

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

= n^{4} \cdot \int cos^{2} (s i x) x d x

Wende U-Substitution an

= n^{4} \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^{4} \frac{1}{s ^{2} i ^{2}} \cdot \int u cos^{2} (u) d u

Wende die partielle Integration an

= n^{4} \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^{4} \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^{4} \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^{4} \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 ^{4}}{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 ^{4}}{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 ^{4}}{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

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