integral from-pi to pi of x*sin(n*x)

1) The math expression:

x \cdot \sin (n x)

contains several variables so the variable: x is chosen for integral computation

Let's integrate:

\int(x \cdot \sin (n x))dx

For

x \cdot \sin (n x)

, we use integration by parts:

\int(u1 \cdot v2')dx = u1 \cdot v2 - \int(u1' \cdot v2)dx

, with u1 =

x

and v2' =

\sin (n x)

Thus,

u1'

1

2) Let's compute the integral v2':

\int(\sin (n x))dx

For the math expression:

\sin (n x)

, we know the derivative formula:

\frac{d}{d x} (\cos (a)) = - \sin (a)

, so we try differentiating with cos

3) Let's perform the differentiation:

\frac{d}{d x} (\cos (n x))

We differentiate using:

\frac{d}{d x} (\cos (f)) = - \frac{d}{d x} (f) \cdot \sin (f)

(where f =

n x

)

- \frac{d}{d x} (n x) \cdot \sin (n x)

- n \cdot \sin (n x)

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

\frac{1}{- n} \cdot \cos (n x)

- \frac{1}{n} \cdot \cos (n x)

- \frac{\cos (n x)}{n}

So the value of

v2

- \frac{\cos (n x)}{n}

4) Let's compute the integral

u1 \cdot v2 - \int(u1' \cdot v2)dx

x \cdot (- \frac{\cos (n x)}{n}) - \int(- \frac{\cos (n x)}{n})dx

For the math expression:

\frac{\cos (n x)}{n}

, we know the derivative formula:

\frac{d}{d x} (\sin (a)) = \cos (a)

, so we try differentiating with sin

5) Let's differentiate:

\frac{d}{d x} (\sin (n x))

Apply the differentiation rule:

\frac{d}{d x} (\sin (f)) = \frac{d}{d x} (f) \cdot \cos (f)

with f =

n x

\frac{d}{d x} (n x) \cdot \cos (n x)

n \cdot \cos (n x)

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

- \frac{x \cdot \cos (n x)}{n} + \frac{1}{n^{2}} \cdot \sin (n x)

▷

- \frac{x \cdot \cos (n x)}{n} + \frac{\sin (n x)}{n^{2}}

The parameter n is real and constant during integration; this antiderivative applies under the condition

n \in ℝ \ {0}

For n = 0, the integrand is identically zero: the integral is 0

6) Let's evaluate for the interval

[- π, π]

(\frac{\sin (n π)}{n^{2}} - \frac{π \cdot \cos (n π)}{n}) - (\frac{\sin (n \cdot (- π))}{n^{2}} - (\frac{- π \cdot \cos (n \cdot (- π))}{n}))

\frac{\sin (n π)}{n^{2}}

- \frac{π \cdot \cos (n π)}{n}

- \frac{\sin (n \cdot (- π))}{n^{2}}

+ \frac{- π \cdot \cos (n \cdot (- π))}{n}

\frac{\sin (n π)}{n^{2}}

- \frac{π \cdot \cos (n π)}{n}

- \frac{\sin (- n π)}{n^{2}}

- \frac{π \cdot \cos (- n π)}{n}

ℹ

\sin (- n π)

- \sin (n π)

‖

\cos (- n π)

\cos (n π)

\frac{\sin (n π)}{n^{2}}

- \frac{π \cdot \cos (n π)}{n}

- (\frac{- \sin (n π)}{n^{2}})

- \frac{π \cdot \cos (n π)}{n}

ℹ

- (\frac{- \sin (n π)}{n^{2}})

\frac{\sin (n π)}{n^{2}}

‖

- \frac{π \cdot \cos (n π)}{n} - \frac{π \cdot \cos (n π)}{n}

-2 \cdot \frac{π \cdot \cos (n π)}{n}

\frac{\sin (n π)}{n^{2}} - 2 \cdot \frac{π \cdot \cos (n π)}{n} + \frac{\sin (n π)}{n^{2}}

ℹ

-2 \cdot \frac{π \cdot \cos (n π)}{n}

\frac{-2 π \cdot \cos (n π)}{n}

‖

\frac{\sin (n π)}{n^{2}} + \frac{\sin (n π)}{n^{2}}

2 \cdot \frac{\sin (n π)}{n^{2}}

2 \cdot \frac{\sin (n π)}{n^{2}} - \frac{2 π \cdot \cos (n π)}{n}

▶Integral:

\frac{2 \cdot \sin (n π)}{n^{2}} - \frac{2 π \cdot \cos (n π)}{n}