What is the integral of x(a-x)*sin((npix)/a) ?

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

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

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

- 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

Solution steps

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

Apply Integration By Parts

: (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

= (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}))

Simplify

(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})

Add a constant to the solution

= - 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

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What is the integral of x(a-x)*sin((npix)/a) ?
The integral of x(a-x)*sin((npix)/a) is