integral of xcos((npix)/4)

解答

\int x cos (\frac{nπ x}{4}) d x

\frac{1}{π ^{2} n ^{2}} (4 πn x sin (\frac{πn x}{4}) + 16 cos (\frac{πn x}{4})) + C

求解步骤

\int x cos (\frac{nπ x}{4}) d x

使用换元积分法

= \int \frac{u cos ( \frac{u}{4} )}{π ^{2} n ^{2}} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{π ^{2} n ^{2}} \cdot \int u cos (\frac{u}{4}) d u

使用分布积分法

= \frac{1}{π ^{2} n ^{2}} (4 u sin (\frac{u}{4}) - \int 4 sin (\frac{u}{4}) d u)

\int 4 sin (\frac{u}{4}) d u = - 16 cos (\frac{u}{4})

= \frac{1}{π ^{2} n ^{2}} (4 u sin (\frac{u}{4}) - (- 16 cos (\frac{u}{4})))

u = nπ x

代回

= \frac{1}{π ^{2} n ^{2}} (4 nπ x sin (\frac{nπ x}{4}) - (- 16 cos (\frac{nπ x}{4})))

化简

= \frac{1}{π ^{2} n ^{2}} (4 πn x sin (\frac{πn x}{4}) + 16 cos (\frac{πn x}{4}))

解答补常数

= \frac{1}{π ^{2} n ^{2}} (4 πn x sin (\frac{πn x}{4}) + 16 cos (\frac{πn x}{4})) + C

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