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

解

\int (1 - x^{2}) cos (nπ x) d x

\frac{1}{πn} sin (πn x) - \frac{1}{π ^{3} n ^{3}} (π^{2} n^{2} x^{2} sin (πn x) - 2 (- πn x cos (πn x) + sin (πn x))) + C

解答ステップ

\int (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 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 cos (πn x) d x - \int x^{2} cos (πn x) d x

\int cos (πn x) d x = \frac{1}{πn} sin (πn x)

\int x^{2} cos (πn x) d x = \frac{1}{π ^{3} n ^{3}} (π^{2} n^{2} x^{2} sin (πn x) - 2 (- πn x cos (πn x) + sin (πn x)))

= \frac{1}{πn} sin (πn x) - \frac{1}{π ^{3} n ^{3}} (π^{2} n^{2} x^{2} sin (πn x) - 2 (- πn x cos (πn x) + sin (πn x)))

定数を解答に追加する

= \frac{1}{πn} sin (πn x) - \frac{1}{π ^{3} n ^{3}} (π^{2} n^{2} x^{2} sin (πn x) - 2 (- πn x cos (πn x) + sin (πn x))) + C