integral from-2 to 5 of 2e^x(5x^2+3)

Lösung

\int_{- 2}^{5} 2 e^{x} (5 x^{2} + 3) d x

2 (88 e^{5} - \frac{53}{e ^{2}})

Dezimale

26106.37046 \dots

Schritte zur Lösung

\int_{- 2}^{5} 2 e^{x} (5 x^{2} + 3) d x

Entferne die Konstante:

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

= 2 \cdot \int_{- 2}^{5} e^{x} (5 x^{2} + 3) d x

Multipliziere mit der Konjugation von

5 x^{2} + 3 : \frac{25 x ^{4} - 9}{5 x ^{2} - 3}

= 2 \cdot \int_{- 2}^{5} e^{x} \frac{25 x ^{4} - 9}{5 x ^{2} - 3} d x

Bestimme unbestimmte (Singularitäts-)Punkte:

x = \frac{3}{5}, x = - \frac{3}{5}

Wenn

b, a < b < c, f (b) =

unbestimmt

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= 2 (\int_{- 2}^{- \frac{3}{5}} e^{x} \frac{25 x ^{4} - 9}{5 x ^{2} - 3} d x + \int_{- \frac{3}{5}}^{\frac{3}{5}} e^{x} \frac{25 x ^{4} - 9}{5 x ^{2} - 3} d x + \int_{\frac{3}{5}}^{5} e^{x} \frac{25 x ^{4} - 9}{5 x ^{2} - 3} d x)

\int_{- 2}^{- \frac{3}{5}} e^{x} \frac{25 x ^{4} - 9}{5 x ^{2} - 3} d x = \frac{16 e ^{2} + 2 15 e ^{2} - 53 e ^{\frac{3}{5}}}{e ^{\frac{3 + 2 5}{5}}}

\int_{- \frac{3}{5}}^{\frac{3}{5}} e^{x} \frac{25 x ^{4} - 9}{5 x ^{2} - 3} d x = 16 e^{\frac{3}{5}} - 10 e^{\frac{3}{5}} \frac{3}{5} + \frac{- 16 - 2 15}{e ^{\frac{3}{5}}}

\int_{\frac{3}{5}}^{5} e^{x} \frac{25 x ^{4} - 9}{5 x ^{2} - 3} d x = 88 e^{5} - 16 e^{\frac{3}{5}} + 10 e^{\frac{3}{5}} \frac{3}{5}

= 2 (\frac{16 e ^{2} + 2 15 e ^{2} - 53 e ^{\frac{3}{5}}}{e ^{\frac{3 + 2 5}{5}}} + 16 e^{\frac{3}{5}} - 10 e^{\frac{3}{5}} \frac{3}{5} + \frac{- 16 - 2 15}{e ^{\frac{3}{5}}} + 88 e^{5} - 16 e^{\frac{3}{5}} + 10 e^{\frac{3}{5}} \frac{3}{5})

Vereinfache

2 (\frac{16 e ^{2} + 2 15 e ^{2} - 53 e ^{\frac{3}{5}}}{e ^{\frac{3 + 2 5}{5}}} + 16 e^{\frac{3}{5}} - 10 e^{\frac{3}{5}} \frac{3}{5} + \frac{- 16 - 2 15}{e ^{\frac{3}{5}}} + 88 e^{5} - 16 e^{\frac{3}{5}} + 10 e^{\frac{3}{5}} \frac{3}{5}) : 2 (88 e^{5} - \frac{53}{e ^{2}})

= 2 (88 e^{5} - \frac{53}{e ^{2}})

\int x 4 + x d x

n or ma l f (x) = x^{3} - 3 x^{2} + 5, at x = - 1

\int_{0}^{6} 2

\int_{1}^{100} \frac{13}{x ^{3}} d x

f (t) = t e^{t}