integral from x^2 to e^x of x(t^2+1)

Lösung

\int_{x^{2}}^{e^{x}} x (t^{2} + 1) d t

x (e^{x} - x^{2} + \frac{e ^{3 x} - x ^{6}}{3})

Schritte zur Lösung

\int_{x^{2}}^{e^{x}} x (t^{2} + 1) d t

Entferne die Konstante:

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

= x \cdot \int_{x^{2}}^{e^{x}} 1 + t^{2} d t

Wende die Summenregel an:

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

= x (\int_{x^{2}}^{e^{x}} 1 d t + \int_{x^{2}}^{e^{x}} t^{2} d t)

\int_{x^{2}}^{e^{x}} 1 d t = e^{x} - x^{2}

\int_{x^{2}}^{e^{x}} t^{2} d t = \frac{e ^{3 x} - x ^{6}}{3}

= x (e^{x} - x^{2} + \frac{e ^{3 x} - x ^{6}}{3})

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