integral of sqrt(1+((e^x-e^{-x))/2)^2}

Lösung

\int 1 + (\frac{e ^{x} - e ^{- x}}{2})^{2} d x

sinh (x) + C

Schritte zur Lösung

\int 1 + (\frac{e ^{x} - e ^{- x}}{2})^{2} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int 1 + (\frac{2 sinh ( x )}{2})^{2} d x

Wende die partielle Integration an

= x cosh^{2} (x) - \int \frac{x cosh ( x ) sinh ( x )}{cosh ^{2} ( x )} d x

\int \frac{x cosh ( x ) sinh ( x )}{cosh ^{2} ( x )} d x = x cosh (x) - sinh (x)

= x cosh^{2} (x) - (x cosh (x) - sinh (x))

Vereinfache

x cosh^{2} (x) - (x cosh (x) - sinh (x)) : sinh (x)

= sinh (x)

Füge eine Konstante zur Lösung hinzu

= sinh (x) + C

\int (7 x - 1) e^{2 x} d x

\int 0.04 x + 27 d x

\int x tan^{2} (x^{2}) sec^{2} (x^{2}) d x

in t e g r a l 3

\int \frac{e ^{2 x} - e ^{3 x}}{e ^{x}} d x