integral from 0 to x of tf(x^2-t^2)

Lösung

\int_{0}^{x} t f (x^{2} - t^{2}) d t

f (\frac{x ^{4}}{2} - \frac{x ^{4}}{4})

Schritte zur Lösung

\int_{0}^{x} t f (x^{2} - t^{2}) d t

Entferne die Konstante:

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

= f \cdot \int_{0}^{x} (x^{2} - t^{2}) t d t

Multipliziere aus

(x^{2} - t^{2}) t : x^{2} t - t^{3}

= f \cdot \int_{0}^{x} x^{2} t - t^{3} 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

= f (\int_{0}^{x} x^{2} t d t - \int_{0}^{x} t^{3} d t)

\int_{0}^{x} x^{2} t d t = \frac{x ^{4}}{2}

\int_{0}^{x} t^{3} d t = \frac{x ^{4}}{4}

= f (\frac{x ^{4}}{2} - \frac{x ^{4}}{4})

\int_{0}^{s i n (x)} ln (t^{3} + 1) d t

\int_{2}^{\infty} x e^{- \frac{x}{2}} d x

\int_{0}^{π} 1 - sin (x) d x

\int_{0}^{1} a x d x

\int_{0}^{t^{6}} u^{7} d u