integral from-1 to 1 of (1-x^2)cos(npix)

解

\int_{- 1}^{1} (1 - x^{2}) cos (nπ x) d x

- \frac{4 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

解答ステップ

\int_{- 1}^{1} (1 - x^{2}) cos (nπ x) d x

拡張

(1 - x^{2}) cos (nπ x) : cos (πn x) - x^{2} cos (πn x)

= \int_{- 1}^{1} cos (πn x) - x^{2} cos (πn x) d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{- 1}^{1} cos (πn x) d x - \int_{- 1}^{1} x^{2} cos (πn x) d x

\int_{- 1}^{1} cos (πn x) d x = 0

\int_{- 1}^{1} x^{2} cos (πn x) d x = \frac{4 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

= 0 - \frac{4 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

簡素化

= - \frac{4 ( - 1 ) ^{n}}{π ^{2} n ^{2}}