integral from 0 to pi of sin^2(tco)s^4t

Soluzione

\int_{0}^{π} sin^{2} (t co) s^{4} t d t

\frac{s ^{4} ( 2 π ^{2} c ^{2} o ^{2} - 2 π co sin ( 2 π co ) - cos ( 2 π co ) + 1 )}{8 c ^{2} o ^{2}}

Fasi della soluzione

\int_{0}^{π} sin^{2} (t co) s^{4} t d t

Porta fuori la costante:

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

= s^{4} \cdot \int_{0}^{π} sin^{2} (co t) t d t

Applicare la sostituzione u

= s^{4} \cdot \int_{0}^{co π} \frac{u sin ^{2} ( u )}{c ^{2} o ^{2}} d u

Porta fuori la costante:

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

= s^{4} \frac{1}{c ^{2} o ^{2}} \cdot \int_{0}^{co π} u sin^{2} (u) d u

Applica l'Integrazione per Parti

= s^{4} \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]_{0}^{π co}

\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))

= s^{4} \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))]_{0}^{π co}

Semplificare

s^{4} \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))]_{0}^{π co} : \frac{s ^{4}}{c ^{2} o ^{2}} [\frac{2 u ^{2} - 2 u sin ( 2 u ) - cos ( 2 u )}{8}]_{0}^{π co}

= \frac{s ^{4}}{c ^{2} o ^{2}} [\frac{2 u ^{2} - 2 u sin ( 2 u ) - cos ( 2 u )}{8}]_{0}^{π co}

Calcola i limiti:

\frac{2 π ^{2} c ^{2} o ^{2} - 2 π co sin ( 2 π co ) - cos ( 2 π co )}{8} + \frac{1}{8}

= \frac{s ^{4}}{c ^{2} o ^{2}} (\frac{2 π ^{2} c ^{2} o ^{2} - 2 π co sin ( 2 π co ) - cos ( 2 π co )}{8} + \frac{1}{8})

Semplificare

= \frac{s ^{4} ( 2 π ^{2} c ^{2} o ^{2} - 2 π co sin ( 2 π co ) - cos ( 2 π co ) + 1 )}{8 c ^{2} o ^{2}}

\frac{d y}{d t} = 0.225 - \frac{y}{200}, y (0) = 90

t a y l or (x^{2} + 1) \cdot e^{x - 1}

im pl i c i t \frac{d y}{d x}, y = x^{\frac{15}{16}}

\int 0.5 e^{x} d x

\int e^{- a x} cos (b x) d x