integral from 0 to 1 of sinh^4(4x)

Lösung

\int_{0}^{1} sinh^{4} (4 x) d x

\frac{\frac{e ^{32} - 2 e ^{24} + 2 e ^{8} - 1}{64 e ^{16}} - \frac{3 ( - 16 e ^{8} + e ^{16} - 1 )}{32 e ^{8}}}{4}

Dezimale

34618.58929 \dots

Schritte zur Lösung

\int_{0}^{1} sinh^{4} (4 x) d x

Wende U-Substitution an

= \int_{0}^{4} sinh^{4} (u) \frac{1}{4} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int_{0}^{4} sinh^{4} (u) d u

Wende integrale Reduktion an

= \frac{1}{4} ([\frac{cosh ( u ) sinh ^{3} ( u )}{4}]_{0}^{4} - \frac{3}{4} \cdot \int_{0}^{4} sinh^{2} (u) d u)

\int_{0}^{4} sinh^{2} (u) d u = \frac{- 16 e ^{8} + e ^{16} - 1}{8 e ^{8}}

= \frac{1}{4} ([\frac{cosh ( u ) sinh ^{3} ( u )}{4}]_{0}^{4} - \frac{3}{4} \cdot \frac{- 16 e ^{8} + e ^{16} - 1}{8 e ^{8}})

Vereinfache

\frac{1}{4} ([\frac{cosh ( u ) sinh ^{3} ( u )}{4}]_{0}^{4} - \frac{3}{4} \cdot \frac{- 16 e ^{8} + e ^{16} - 1}{8 e ^{8}}) : \frac{1}{4} ([\frac{1}{4} sinh^{3} (u) cosh (u)]_{0}^{4} - \frac{3 ( - 16 e ^{8} + e ^{16} - 1 )}{32 e ^{8}})

= \frac{1}{4} ([\frac{1}{4} sinh^{3} (u) cosh (u)]_{0}^{4} - \frac{3 ( - 16 e ^{8} + e ^{16} - 1 )}{32 e ^{8}})

Berechne die Grenzen:

\frac{( e ^{8} - 1 ) ^{3} ( e ^{8} + 1 )}{64 e ^{16}}

= \frac{1}{4} (\frac{( e ^{8} - 1 ) ^{3} ( e ^{8} + 1 )}{64 e ^{16}} - \frac{3 ( - 16 e ^{8} + e ^{16} - 1 )}{32 e ^{8}})

Vereinfache

= \frac{\frac{e ^{32} - 2 e ^{24} + 2 e ^{8} - 1}{64 e ^{16}} - \frac{3 ( - 16 e ^{8} + e ^{16} - 1 )}{32 e ^{8}}}{4}

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