What is the integral of-xcos((nxpi)/l) ?

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

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integral of-xcos((nxpi)/l)

\int - x cos (\frac{n x π}{l}) d x

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

Solution steps

\int - x cos (\frac{n x π}{l}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int x cos (\frac{n x π}{l}) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{d}{d x} (4 x^{3} - 3)

(e^{t} + e^{- t}) \frac{d x}{d t} = x^{2}

x^{2} y^{^{''}} + x y^{^{'}} + y = 0

\int 5 sec^{3} (x) d x

\frac{d}{d x} (e^{2 x} sin (2 x))

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