integral of (5x^2+5x+3)^4(2x+1)

Lösung

\int (5 x^{2} + 5 x + 3)^{4} (2 x + 1) d x

125 x^{10} + 625 x^{9} + 1625 x^{8} + 2750 x^{7} + 3325 x^{6} + 2975 x^{5} + 1995 x^{4} + 990 x^{3} + 351 x^{2} + 81 x + C

Schritte zur Lösung

\int (5 x^{2} + 5 x + 3)^{4} (2 x + 1) d x

Multipliziere aus

(5 x^{2} + 5 x + 3)^{4} (2 x + 1) : 1250 x^{9} + 5625 x^{8} + 13000 x^{7} + 19250 x^{6} + 19950 x^{5} + 14875 x^{4} + 7980 x^{3} + 2970 x^{2} + 702 x + 81

= \int 1250 x^{9} + 5625 x^{8} + 13000 x^{7} + 19250 x^{6} + 19950 x^{5} + 14875 x^{4} + 7980 x^{3} + 2970 x^{2} + 702 x + 81 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 1250 x^{9} d x + \int 5625 x^{8} d x + \int 13000 x^{7} d x + \int 19250 x^{6} d x + \int 19950 x^{5} d x + \int 14875 x^{4} d x + \int 7980 x^{3} d x + \int 2970 x^{2} d x + \int 702 x d x + \int 81 d x

\int 1250 x^{9} d x = 125 x^{10}

\int 5625 x^{8} d x = 625 x^{9}

\int 13000 x^{7} d x = 1625 x^{8}

\int 19250 x^{6} d x = 2750 x^{7}

\int 19950 x^{5} d x = 3325 x^{6}

\int 14875 x^{4} d x = 2975 x^{5}

\int 7980 x^{3} d x = 1995 x^{4}

\int 2970 x^{2} d x = 990 x^{3}

\int 702 x d x = 351 x^{2}

\int 81 d x = 81 x

= 125 x^{10} + 625 x^{9} + 1625 x^{8} + 2750 x^{7} + 3325 x^{6} + 2975 x^{5} + 1995 x^{4} + 990 x^{3} + 351 x^{2} + 81 x

Füge eine Konstante zur Lösung hinzu

= 125 x^{10} + 625 x^{9} + 1625 x^{8} + 2750 x^{7} + 3325 x^{6} + 2975 x^{5} + 1995 x^{4} + 990 x^{3} + 351 x^{2} + 81 x + C

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