integral from 0 to 1 of 3x(x^2+1)^3

Soluzione

\int_{0}^{1} 3 x (x^{2} + 1)^{3} d x

\frac{45}{8}

Notazione decimale

5.625

Fasi della soluzione

\int_{0}^{1} 3 x (x^{2} + 1)^{3} d x

Porta fuori la costante:

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

= 3 \cdot \int_{0}^{1} x (x^{2} + 1)^{3} d x

Espandi

x (x^{2} + 1)^{3} : x^{7} + 3 x^{5} + 3 x^{3} + x

= 3 \cdot \int_{0}^{1} x^{7} + 3 x^{5} + 3 x^{3} + x d x

Applica la Regola della Somma:

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

= 3 (\int_{0}^{1} x^{7} d x + \int_{0}^{1} 3 x^{5} d x + \int_{0}^{1} 3 x^{3} d x + \int_{0}^{1} x d x)

\int_{0}^{1} x^{7} d x = \frac{1}{8}

\int_{0}^{1} 3 x^{5} d x = \frac{1}{2}

\int_{0}^{1} 3 x^{3} d x = \frac{3}{4}

\int_{0}^{1} x d x = \frac{1}{2}

= 3 (\frac{1}{8} + \frac{1}{2} + \frac{3}{4} + \frac{1}{2})

Semplificare

3 (\frac{1}{8} + \frac{1}{2} + \frac{3}{4} + \frac{1}{2}) : \frac{45}{8}

= \frac{45}{8}

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

\int_{- π}^{π} x^{4} cos (n x) d x

\int_{0}^{0.5} e^{- x} d x

\int_{0}^{6} (4 x - 1)^{2} d x

\int_{3}^{9} ∣ sin (x) ∣ d x