integral of 7sin^2(xco)s^3x

Lösung

\int 7 sin^{2} (x co) s^{3} x d x

\frac{7 s ^{3}}{8 c ^{2} o ^{2}} (2 c^{2} o^{2} x^{2} - 2 co x sin (2 co x) - cos (2 co x)) + C

Schritte zur Lösung

\int 7 sin^{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

= 7 s^{3} \cdot \int sin^{2} (co x) x d x

Wende U-Substitution an

= 7 s^{3} \cdot \int \frac{u sin ^{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

= 7 s^{3} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{2} (u) d u

Wende die partielle Integration an

= 7 s^{3} \frac{1}{c ^{2} o ^{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))

= 7 s^{3} \frac{1}{c ^{2} o ^{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 = co x

ein

= 7 s^{3} \frac{1}{c ^{2} o ^{2}} (\frac{1}{2} co x (co x - \frac{1}{2} sin (2 co x)) - \frac{1}{2} (\frac{( co x ) ^{2}}{2} + \frac{1}{4} cos (2 co x)))

Vereinfache

7 s^{3} \frac{1}{c ^{2} o ^{2}} (\frac{1}{2} co x (co x - \frac{1}{2} sin (2 co x)) - \frac{1}{2} (\frac{( co x ) ^{2}}{2} + \frac{1}{4} cos (2 co x))) : \frac{7 s ^{3}}{8 c ^{2} o ^{2}} (2 c^{2} o^{2} x^{2} - 2 co x sin (2 co x) - cos (2 co x))

= \frac{7 s ^{3}}{8 c ^{2} o ^{2}} (2 c^{2} o^{2} x^{2} - 2 co x sin (2 co x) - cos (2 co x))

Füge eine Konstante zur Lösung hinzu

= \frac{7 s ^{3}}{8 c ^{2} o ^{2}} (2 c^{2} o^{2} x^{2} - 2 co x sin (2 co x) - cos (2 co x)) + C

\int_{- 3}^{3} \frac{12}{x ^{2} - 2 x - 35} d x

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