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Beliebt Rechnen >

integral of (mt^4+n)^5

Lösung

\int (m t^{4} + n)^{5} d t

Lösung

\frac{m ^{5} t ^{21}}{21} + \frac{5 m ^{4} n t ^{17}}{17} + \frac{10 m ^{3} n ^{2} t ^{13}}{13} + \frac{10 m ^{2} n ^{3} t ^{9}}{9} + m n^{4} t^{5} + n^{5} t + C

Schritte zur Lösung

\int (m t^{4} + n)^{5} d t

Multipliziere aus

(m t^{4} + n)^{5} : m^{5} t^{20} + 5 m^{4} n t^{16} + 10 m^{3} n^{2} t^{12} + 10 m^{2} n^{3} t^{8} + 5 m n^{4} t^{4} + n^{5}

= \int m^{5} t^{20} + 5 m^{4} n t^{16} + 10 m^{3} n^{2} t^{12} + 10 m^{2} n^{3} t^{8} + 5 m n^{4} t^{4} + n^{5} d t

Wende die Summenregel an:

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

= \int m^{5} t^{20} d t + \int 5 m^{4} n t^{16} d t + \int 10 m^{3} n^{2} t^{12} d t + \int 10 m^{2} n^{3} t^{8} d t + \int 5 m n^{4} t^{4} d t + \int n^{5} d t

\int m^{5} t^{20} d t = \frac{m ^{5} t ^{21}}{21}

\int 5 m^{4} n t^{16} d t = \frac{5 m ^{4} n t ^{17}}{17}

\int 10 m^{3} n^{2} t^{12} d t = \frac{10 m ^{3} n ^{2} t ^{13}}{13}

\int 10 m^{2} n^{3} t^{8} d t = \frac{10 m ^{2} n ^{3} t ^{9}}{9}

\int 5 m n^{4} t^{4} d t = m n^{4} t^{5}

\int n^{5} d t = n^{5} t

= \frac{m ^{5} t ^{21}}{21} + \frac{5 m ^{4} n t ^{17}}{17} + \frac{10 m ^{3} n ^{2} t ^{13}}{13} + \frac{10 m ^{2} n ^{3} t ^{9}}{9} + m n^{4} t^{5} + n^{5} t

Füge eine Konstante zur Lösung hinzu

= \frac{m ^{5} t ^{21}}{21} + \frac{5 m ^{4} n t ^{17}}{17} + \frac{10 m ^{3} n ^{2} t ^{13}}{13} + \frac{10 m ^{2} n ^{3} t ^{9}}{9} + m n^{4} t^{5} + n^{5} t + C

Graph

Beliebte Beispiele

integral of 2tan(x)sec^5(x)

\int 2 tan (x) sec^{5} (x) d x

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

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\int \frac{5}{4} 5 z + 7 d z

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\int x^{3} x^{4} + 8 d x