integral from 0 to x^2 of cos(t^3)

Lösung

\int_{0}^{x^{2}} cos (t^{3}) d t

- \frac{x ^{2} ( 3 i ( x ^{2} ) ^{3} Γ ( \frac{1}{3} , - i ( x ^{2} ) ^{3} ) + 3 - i ( x ^{2} ) ^{3} Γ ( \frac{1}{3} , i ( x ^{2} ) ^{3} ) )}{6 3 ( x ^{2} ) ^{6}} - - \frac{0 \cdot ( 3 i 0 ^{3} Γ ( \frac{1}{3} , - i 0 ^{3} ) + 3 - i 0 ^{3} Γ ( \frac{1}{3} , i 0 ^{3} ) )}{6 3 0 ^{6}}

Schritte zur Lösung

\int_{0}^{x^{2}} cos (t^{3}) d t

Das ist ein nicht elementares Integral :

\int cos (x^{3}) d x = - \frac{x ( 3 i x ^{3} Γ ( \frac{1}{3} , - i x ^{3} ) + 3 - i x ^{3} Γ ( \frac{1}{3} , i x ^{3} ) )}{6 3 x ^{6}}

= - \frac{t ( 3 i t ^{3} Γ ( \frac{1}{3} , - i t ^{3} ) + 3 - i t ^{3} Γ ( \frac{1}{3} , i t ^{3} ) )}{6 3 t ^{6}}_{0}^{x^{2}}

Berechne die Grenzen:

- \frac{x ^{2} ( 3 i ( x ^{2} ) ^{3} Γ ( \frac{1}{3} , - i ( x ^{2} ) ^{3} ) + 3 - i ( x ^{2} ) ^{3} Γ ( \frac{1}{3} , i ( x ^{2} ) ^{3} ) )}{6 3 ( x ^{2} ) ^{6}} - - \frac{0 \cdot ( 3 i 0 ^{3} Γ ( \frac{1}{3} , - i 0 ^{3} ) + 3 - i 0 ^{3} Γ ( \frac{1}{3} , i 0 ^{3} ) )}{6 3 0 ^{6}}

= - \frac{x ^{2} ( 3 i ( x ^{2} ) ^{3} Γ ( \frac{1}{3} , - i ( x ^{2} ) ^{3} ) + 3 - i ( x ^{2} ) ^{3} Γ ( \frac{1}{3} , i ( x ^{2} ) ^{3} ) )}{6 3 ( x ^{2} ) ^{6}} - - \frac{0 \cdot ( 3 i 0 ^{3} Γ ( \frac{1}{3} , - i 0 ^{3} ) + 3 - i 0 ^{3} Γ ( \frac{1}{3} , i 0 ^{3} ) )}{6 3 0 ^{6}}

\int_{0}^{\frac{π}{3}} 2 cos (x) d x

\int_{0}^{2} 5^{3 x} d x

\int_{0}^{2} (2 x^{3} - 2 x) d x

\int_{1}^{3} 2^{5 x} d x

\int_{0}^{3} \frac{2 x}{( x ^{2} + 3 ) ^{2}} d x