de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of x(a-x)*sin((npix)/a)

Lösung

\int x (a - x) \cdot sin (\frac{nπ x}{a}) d x

Lösung

- a^{2} \frac{1}{πn} x cos (\frac{πn x}{a}) + a^{2} \frac{a}{π ^{2} n ^{2}} sin (\frac{πn x}{a}) + a \frac{1}{πn} x^{2} cos (\frac{πn x}{a}) - 2 a \frac{a}{π ^{2} n ^{2}} x sin (\frac{πn x}{a}) - 2 a \frac{a ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{a}) + C

Schritte zur Lösung

\int x (a - x) sin (\frac{nπ x}{a}) d x

Wende die partielle Integration an

= (a - x) (- a \frac{1}{πn} x cos (\frac{πn x}{a}) + a \frac{a}{π ^{2} n ^{2}} sin (\frac{πn x}{a})) - \int a \frac{1}{πn} x cos (\frac{πn x}{a}) - a \frac{a}{π ^{2} n ^{2}} sin (\frac{πn x}{a}) d x

\int a \frac{1}{πn} x cos (\frac{πn x}{a}) - a \frac{a}{π ^{2} n ^{2}} sin (\frac{πn x}{a}) d x = a \frac{a}{π ^{2} n ^{2}} x sin (\frac{πn x}{a}) + 2 a \frac{a ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{a})

= (a - x) (- a \frac{1}{πn} x cos (\frac{πn x}{a}) + a \frac{a}{π ^{2} n ^{2}} sin (\frac{πn x}{a})) - (a \frac{a}{π ^{2} n ^{2}} x sin (\frac{πn x}{a}) + 2 a \frac{a ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{a}))

Vereinfache

(a - x) (- a \frac{1}{πn} x cos (\frac{πn x}{a}) + a \frac{a}{π ^{2} n ^{2}} sin (\frac{πn x}{a})) - (a \frac{a}{π ^{2} n ^{2}} x sin (\frac{πn x}{a}) + 2 a \frac{a ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{a})) : - a^{2} \frac{1}{πn} x cos (\frac{πn x}{a}) + a^{2} \frac{a}{π ^{2} n ^{2}} sin (\frac{πn x}{a}) + a \frac{1}{πn} x^{2} cos (\frac{πn x}{a}) - 2 a \frac{a}{π ^{2} n ^{2}} x sin (\frac{πn x}{a}) - 2 a \frac{a ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{a})

= - a^{2} \frac{1}{πn} x cos (\frac{πn x}{a}) + a^{2} \frac{a}{π ^{2} n ^{2}} sin (\frac{πn x}{a}) + a \frac{1}{πn} x^{2} cos (\frac{πn x}{a}) - 2 a \frac{a}{π ^{2} n ^{2}} x sin (\frac{πn x}{a}) - 2 a \frac{a ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{a})

Füge eine Konstante zur Lösung hinzu

= - a^{2} \frac{1}{πn} x cos (\frac{πn x}{a}) + a^{2} \frac{a}{π ^{2} n ^{2}} sin (\frac{πn x}{a}) + a \frac{1}{πn} x^{2} cos (\frac{πn x}{a}) - 2 a \frac{a}{π ^{2} n ^{2}} x sin (\frac{πn x}{a}) - 2 a \frac{a ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{a}) + C

Graph

Beliebte Beispiele

integral of xe^{5x^2+9}

\int x e^{5 x^{2} + 9} d x

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

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

integral of 50-3xe^{0.01x^2}

\int 50 - 3 x e^{0.01 x^{2}} d x

integral of (21t^2-10t)/(7t^3-5t^2+13)

\int \frac{21 t ^{2} - 10 t}{7 t ^{3} - 5 t ^{2} + 13} d t

integral of (x^2+2)/(sqrt(x^3+6x))

\int \frac{x ^{2} + 2}{x ^{3} + 6 x} d x