integral of [(3x^5-4)^2*5x^4]

Lösung

\int [(3 x^{5} - 4)^{2} \cdot 5 x^{4}] d x

3 x^{15} - 12 x^{10} + 16 x^{5} + C

Schritte zur Lösung

\int (3 x^{5} - 4)^{2} \cdot 5 x^{4} d x

Entferne die Konstante:

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

= 5 \cdot \int (3 x^{5} - 4)^{2} x^{4} d x

Multipliziere aus

(3 x^{5} - 4)^{2} x^{4} : 9 x^{14} - 24 x^{9} + 16 x^{4}

= 5 \cdot \int 9 x^{14} - 24 x^{9} + 16 x^{4} 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

= 5 (\int 9 x^{14} d x - \int 24 x^{9} d x + \int 16 x^{4} d x)

\int 9 x^{14} d x = \frac{3 x ^{15}}{5}

\int 24 x^{9} d x = \frac{12 x ^{10}}{5}

\int 16 x^{4} d x = \frac{16 x ^{5}}{5}

= 5 (\frac{3 x ^{15}}{5} - \frac{12 x ^{10}}{5} + \frac{16 x ^{5}}{5})

Vereinfache

5 (\frac{3 x ^{15}}{5} - \frac{12 x ^{10}}{5} + \frac{16 x ^{5}}{5}) : 3 x^{15} - 12 x^{10} + 16 x^{5}

= 3 x^{15} - 12 x^{10} + 16 x^{5}

Füge eine Konstante zur Lösung hinzu

= 3 x^{15} - 12 x^{10} + 16 x^{5} + C

\int_{0}^{1} π (3 x^{2} - 3 x^{6}) d x

\frac{\partial}{\partial x} (8 - x^{2} - y^{2})

\int \frac{1}{y ^{2} - 4} d y

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

x \to - 2 lim (\frac{x ^{n} - 1}{x - 1})