integral from-1 to 0 of (x+1)cos(npix)

解答

\int_{- 1}^{0} (x + 1) cos (nπ x) d x

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

求解步骤

\int_{- 1}^{0} (x + 1) cos (nπ x) d x

乘开

(x + 1) cos (nπ x) : x cos (πn x) + cos (πn x)

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

使用积分加法定则:

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

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

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

\int_{- 1}^{0} cos (πn x) d x = 0

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

化简

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

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