What is the integral from-1 to 1 of |x|cos(npix) ?

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

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integral from-1 to 1 of |x|cos(npix)

\int_{- 1}^{1} ∣ x ∣ cos (nπ x) d x

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

Solution steps

\int_{- 1}^{1} ∣ x ∣ cos (nπ x) d x

Eliminate Absolutes

= \int_{- 1}^{0} (- x) cos (nπ x) d x + \int_{0}^{1} x cos (nπ x) d x

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

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

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

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