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)

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}}

\int_{- 1}^{0} x^{3} (x^{2} - 1)^{10} d x

\int_{- 2}^{0} 2 x^{3} + 6 x^{2} + 4 x d x

\int_{0}^{8} (x - \frac{1}{2}) d x

\int_{0}^{1} x^{2} - 3 x + 5 d x

\int_{0}^{1} x^{2} - 3 x + 6 d x

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)