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Beliebt Rechnen >

integral of (2x^5+3x^2)/(sqrt(2x^3+1))

Lösung

\int \frac{2 x ^{5} + 3 x ^{2}}{2 x ^{3} + 1} d x

Lösung

\frac{1}{3} (\frac{1}{3} (2 x^{3} + 1)^{\frac{3}{2}} + 2 2 x^{3} + 1) + C

Schritte zur Lösung

\int \frac{2 x ^{5} + 3 x ^{2}}{2 x ^{3} + 1} d x

Wende U-Substitution an

= \int \frac{u ^{2} + 2}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int u^{2} + 2 d u

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 u^{2} d u + \int 2 d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 2 d u = 2 u

= \frac{1}{3} (\frac{u ^{3}}{3} + 2 u)

Setze in

u = 2 x^{3} + 1

ein

= \frac{1}{3} (\frac{( 2 x ^{3} + 1 ) ^{3}}{3} + 2 2 x^{3} + 1)

Vereinfache

\frac{1}{3} (\frac{( 2 x ^{3} + 1 ) ^{3}}{3} + 2 2 x^{3} + 1) : \frac{1}{3} (\frac{1}{3} (2 x^{3} + 1)^{\frac{3}{2}} + 2 2 x^{3} + 1)

= \frac{1}{3} (\frac{1}{3} (2 x^{3} + 1)^{\frac{3}{2}} + 2 2 x^{3} + 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (\frac{1}{3} (2 x^{3} + 1)^{\frac{3}{2}} + 2 2 x^{3} + 1) + C

Graph

Beliebte Beispiele

integral of ((2))/(sqrt(3+2x))

\int \frac{( 2 )}{3 + 2 x} d x

integral of (1/(1+x))^2

\int (\frac{1}{1 + x})^{2} d x

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\int \frac{1}{1} d x

integral of (1-z)/(z^2+2z+1)

\int \frac{1 - z}{z ^{2} + 2 z + 1} d z

integral of (3x)/((x+5)^2)

\int \frac{3 x}{( x + 5 ) ^{2}} d x