What is the integral of xsin((npix)/l) ?

The integral of xsin((npix)/l) is -l 1/(pin)xcos((pinx)/l)+l l/(pi^2n^2)sin((pinx)/l)+C

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integral of xsin((npix)/l)

\int x sin (\frac{nπ x}{l}) d x

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

Solution steps

\int x sin (\frac{nπ x}{l}) d x

Apply Integration By Parts

: - l \frac{1}{πn} x cos (\frac{πn x}{l}) - \int - l \frac{1}{πn} cos (\frac{πn x}{l}) d x

= - l \frac{1}{πn} x cos (\frac{πn x}{l}) - \int - l \frac{1}{πn} cos (\frac{πn x}{l}) d x

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

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

Simplify

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

Add a constant to the solution

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

f (x) = x^{2} + 8, (- 2, 1)

\int_{- 1}^{4} 3 x^{2} - x d x

\int \frac{20}{1 - cos ( 5 x )} d x

x \to \frac{π}{6} lim (cos (x))

t \to 1 lim (t + 2)

What is the integral of xsin((npix)/l) ?
The integral of xsin((npix)/l) is -l 1/(pin)xcos((pinx)/l)+l l/(pi^2n^2)sin((pinx)/l)+C