integral from 1 to 3 of sin(x^3+2)

Lösung

\int_{1}^{3} sin (x^{3} + 2) d x

sin (2) - \frac{3 ( 3 i 3 ^{3} Γ ( \frac{1}{3} , - i 3 ^{3} ) + 3 - i 3 ^{3} Γ ( \frac{1}{3} , i 3 ^{3} ) )}{6 3 3 ^{6}} - - \frac{1 \cdot ( 3 i 1 ^{3} Γ ( \frac{1}{3} , - i 1 ^{3} ) + 3 - i 1 ^{3} Γ ( \frac{1}{3} , i 1 ^{3} ) )}{6 3 1 ^{6}} + cos (2) (- \frac{1}{2} i (\frac{3Γ ( \frac{1}{3} , i 3 ^{3} )}{3 3 i 3 ^{3}} - \frac{3Γ ( \frac{1}{3} , - i 3 ^{3} )}{3 3 - i 3 ^{3}}) - (- \frac{1}{2} i (\frac{1 \cdot Γ ( \frac{1}{3} , i 1 ^{3} )}{3 3 i 1 ^{3}} - \frac{1 \cdot Γ ( \frac{1}{3} , - i 1 ^{3} )}{3 3 - i 1 ^{3}})))

Schritte zur Lösung

\int_{1}^{3} sin (x^{3} + 2) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{1}^{3} cos (x^{3}) sin (2) + cos (2) sin (x^{3}) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{3} cos (x^{3}) sin (2) d x + \int_{1}^{3} cos (2) sin (x^{3}) d x

\int_{1}^{3} cos (x^{3}) sin (2) d x = sin (2) - \frac{3 ( 3 i 3 ^{3} Γ ( \frac{1}{3} , - i 3 ^{3} ) + 3 - i 3 ^{3} Γ ( \frac{1}{3} , i 3 ^{3} ) )}{6 3 3 ^{6}} - - \frac{1 \cdot ( 3 i 1 ^{3} Γ ( \frac{1}{3} , - i 1 ^{3} ) + 3 - i 1 ^{3} Γ ( \frac{1}{3} , i 1 ^{3} ) )}{6 3 1 ^{6}}

\int_{1}^{3} cos (2) sin (x^{3}) d x = cos (2) (- \frac{1}{2} i (\frac{3Γ ( \frac{1}{3} , i 3 ^{3} )}{3 3 i 3 ^{3}} - \frac{3Γ ( \frac{1}{3} , - i 3 ^{3} )}{3 3 - i 3 ^{3}}) - (- \frac{1}{2} i (\frac{1 \cdot Γ ( \frac{1}{3} , i 1 ^{3} )}{3 3 i 1 ^{3}} - \frac{1 \cdot Γ ( \frac{1}{3} , - i 1 ^{3} )}{3 3 - i 1 ^{3}})))

= sin (2) - \frac{3 ( 3 i 3 ^{3} Γ ( \frac{1}{3} , - i 3 ^{3} ) + 3 - i 3 ^{3} Γ ( \frac{1}{3} , i 3 ^{3} ) )}{6 3 3 ^{6}} - - \frac{1 \cdot ( 3 i 1 ^{3} Γ ( \frac{1}{3} , - i 1 ^{3} ) + 3 - i 1 ^{3} Γ ( \frac{1}{3} , i 1 ^{3} ) )}{6 3 1 ^{6}} + cos (2) (- \frac{1}{2} i (\frac{3Γ ( \frac{1}{3} , i 3 ^{3} )}{3 3 i 3 ^{3}} - \frac{3Γ ( \frac{1}{3} , - i 3 ^{3} )}{3 3 - i 3 ^{3}}) - (- \frac{1}{2} i (\frac{1 \cdot Γ ( \frac{1}{3} , i 1 ^{3} )}{3 3 i 1 ^{3}} - \frac{1 \cdot Γ ( \frac{1}{3} , - i 1 ^{3} )}{3 3 - i 1 ^{3}})))

\frac{d x}{d t} = 1.5 t - \frac{2 x}{100 + t}

\frac{d}{d t} (e^{t} + 4)

a re a \frac{3}{2} x^{2}, - \frac{1}{2} x^{2} + 2

\int \frac{7 x}{36 x ^{2} + 1} d x

a re a x = 10 - 2 y, x = y^{2} + 2