integral from 0 to 1/2 of cosh(x^2)

Lösung

\int_{0}^{\frac{1}{2}} cosh (x^{2}) d x

0.5 (0.46128 \dots + 0.88622 \dots erfi (\frac{1}{2}))

Schritte zur Lösung

\int_{0}^{\frac{1}{2}} cosh (x^{2}) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{\frac{1}{2}} \frac{e ^{- x^{2}} + e ^{x^{2}}}{2} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{1}{2}} e^{- x^{2}} + e^{x^{2}} 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

= \frac{1}{2} (\int_{0}^{\frac{1}{2}} e^{- x^{2}} d x + \int_{0}^{\frac{1}{2}} e^{x^{2}} d x)

\int_{0}^{\frac{1}{2}} e^{- x^{2}} d x = 0.46128 \dots

\int_{0}^{\frac{1}{2}} e^{x^{2}} d x = \frac{π}{2} erfi (\frac{1}{2})

= \frac{1}{2} (0.46128 \dots + \frac{π}{2} erfi (\frac{1}{2}))

Vereinfache

= 0.5 (0.46128 \dots + 0.88622 \dots erfi (\frac{1}{2}))

\int_{0}^{π} e^{- x} cos (x) d x

\int_{0}^{2} (x + 1)^{6} d x

\int_{0}^{0.1} arctan (x^{2}) d x

\int_{- 3}^{2} x^{2} - 4 d x

\int_{\frac{1}{4}}^{\frac{1}{2}} 2 x^{3} d x