What is the integral from-pi to pi of |x|cos(nx) ?

The integral from-pi to pi of |x|cos(nx) is -(1-(-1)^n)/(n^2)+((-1)^n-1)/(n^2)

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integral from-pi to pi of |x|cos(nx)

\int_{- π}^{π} ∣ x ∣ cos (n x) d x

- \frac{1 - ( - 1 ) ^{n}}{n ^{2}} + \frac{( - 1 ) ^{n} - 1}{n ^{2}}

Solution steps

\int_{- π}^{π} ∣ x ∣ cos (n x) d x

Eliminate Absolutes

= \int_{- π}^{0} (- x) cos (n x) d x + \int_{0}^{π} x cos (n x) d x

\int_{- π}^{0} (- x) cos (n x) d x = - \frac{1 - ( - 1 ) ^{n}}{n ^{2}}

\int_{0}^{π} x cos (n x) d x = \frac{( - 1 ) ^{n} - 1}{n ^{2}}

= - \frac{1 - ( - 1 ) ^{n}}{n ^{2}} + \frac{( - 1 ) ^{n} - 1}{n ^{2}}

\int_{0}^{1} \frac{x ^{4} + 4}{x} d x

x \to 4 lim (\frac{1}{( 4 - x ) ^{2}})

\frac{d}{d x} (a + b x cos (3 x) + c x sin (3 x))

\frac{\partial}{\partial x} (2 x y - 2 y)

\frac{\partial}{\partial y} (\frac{1}{1 + x ^{2} + y ^{2}})

What is the integral from-pi to pi of |x|cos(nx) ?
The integral from-pi to pi of |x|cos(nx) is -(1-(-1)^n)/(n^2)+((-1)^n-1)/(n^2)