integral of (x^3)/(sqrt(x^2+49))

Lösung

\int \frac{x ^{3}}{x ^{2} + 49} d x

\frac{1}{3} (x^{2} + 49)^{\frac{3}{2}} - 49 x^{2} + 49 + C

Schritte zur Lösung

\int \frac{x ^{3}}{x ^{2} + 49} d x

Wende U-Substitution an

= \int u^{2} - 49 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

= \int u^{2} d u - \int 49 d u

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

\int 49 d u = 49 u

= \frac{u ^{3}}{3} - 49 u

Setze in

u = x^{2} + 49

ein

= \frac{( x ^{2} + 49 ) ^{3}}{3} - 49 x^{2} + 49

Vereinfache

\frac{( x ^{2} + 49 ) ^{3}}{3} - 49 x^{2} + 49 : \frac{1}{3} (x^{2} + 49)^{\frac{3}{2}} - 49 x^{2} + 49

= \frac{1}{3} (x^{2} + 49)^{\frac{3}{2}} - 49 x^{2} + 49

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (x^{2} + 49)^{\frac{3}{2}} - 49 x^{2} + 49 + C

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

t a y l or e^{- h} + e^{h}

d er i v a t i v e 20 (1 + e^{10 - t})^{- 1}

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\frac{\partial}{\partial x} (x^{4} + y^{2} - 4)