integral x^5(x^2+3)^4

Lösung

integral

x^{5} (x^{2} + 3)^{4}

\frac{x ^{14}}{14} + x^{12} + \frac{27 x ^{10}}{5} + \frac{27 x ^{8}}{2} + \frac{27 x ^{6}}{2} + C

Schritte zur Lösung

\int x^{5} (x^{2} + 3)^{4} d x

Multipliziere aus

x^{5} (x^{2} + 3)^{4} : x^{13} + 12 x^{11} + 54 x^{9} + 108 x^{7} + 81 x^{5}

= \int x^{13} + 12 x^{11} + 54 x^{9} + 108 x^{7} + 81 x^{5} 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

= \int x^{13} d x + \int 12 x^{11} d x + \int 54 x^{9} d x + \int 108 x^{7} d x + \int 81 x^{5} d x

\int x^{13} d x = \frac{x ^{14}}{14}

\int 12 x^{11} d x = x^{12}

\int 54 x^{9} d x = \frac{27 x ^{10}}{5}

\int 108 x^{7} d x = \frac{27 x ^{8}}{2}

\int 81 x^{5} d x = \frac{27 x ^{6}}{2}

= \frac{x ^{14}}{14} + x^{12} + \frac{27 x ^{10}}{5} + \frac{27 x ^{8}}{2} + \frac{27 x ^{6}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{14}}{14} + x^{12} + \frac{27 x ^{10}}{5} + \frac{27 x ^{8}}{2} + \frac{27 x ^{6}}{2} + C

in t e g r a l (4 + x)^{\frac{1}{2}}

in t e g r a l x^{2} + 2 x + 3

in t e g r a l \frac{s}{( s ^{2} + 1 )}

in t e g r a l x^{1}

in t e g r a l \frac{x ^{4}}{( x ^{3} + 1 )}