integral from 1 to 4 of (x^5-x)/(3x^3)

Lösung

\int_{1}^{4} \frac{x ^{5} - x}{3 x ^{3}} d x

\frac{27}{4}

Dezimale

6.75

Schritte zur Lösung

\int_{1}^{4} \frac{x ^{5} - x}{3 x ^{3}} d x

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int_{1}^{4} \frac{x ^{5} - x}{x ^{3}} d x

Multipliziere aus

\frac{x ^{5} - x}{x ^{3}} : x^{2} - \frac{1}{x ^{2}}

= \frac{1}{3} \cdot \int_{1}^{4} x^{2} - \frac{1}{x ^{2}} 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

= \frac{1}{3} (\int_{1}^{4} x^{2} d x - \int_{1}^{4} \frac{1}{x ^{2}} d x)

\int_{1}^{4} x^{2} d x = 21

\int_{1}^{4} \frac{1}{x ^{2}} d x = \frac{3}{4}

= \frac{1}{3} (21 - \frac{3}{4})

Vereinfache

\frac{1}{3} (21 - \frac{3}{4}) : \frac{27}{4}

= \frac{27}{4}

\int_{1}^{2} \frac{3}{7} x^{3} d x

\int_{0}^{2 π} (sin (4 x))^{2} d x

\int_{4}^{6} x^{2} x - 4 d x

\int_{2}^{4} (x y^{2} + x^{2} + y + 4) d y

\int_{- 5}^{- 3} x^{2} - 4 d x