integral of 2x^2cos((npix)/2)

解答

\int 2 x^{2} cos (\frac{nπ x}{2}) d x

2 (\frac{2}{πn} x^{2} sin (\frac{πn x}{2}) - \frac{16}{π ^{3} n ^{3}} (- \frac{πn}{2} x cos (\frac{πn x}{2}) + sin (\frac{πn x}{2}))) + C

求解步骤

\int 2 x^{2} cos (\frac{nπ x}{2}) d x

提出常数:

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

= 2 \cdot \int x^{2} cos (\frac{nπ x}{2}) d x

使用分布积分法

= 2 (\frac{2}{πn} x^{2} sin (\frac{πn x}{2}) - \int \frac{4}{πn} x sin (\frac{πn x}{2}) d x)

\int \frac{4}{πn} x sin (\frac{πn x}{2}) d x = \frac{16}{π ^{3} n ^{3}} (- \frac{πn}{2} x cos (\frac{πn x}{2}) + sin (\frac{πn x}{2}))

= 2 (\frac{2}{πn} x^{2} sin (\frac{πn x}{2}) - \frac{16}{π ^{3} n ^{3}} (- \frac{πn}{2} x cos (\frac{πn x}{2}) + sin (\frac{πn x}{2})))

解答补常数

= 2 (\frac{2}{πn} x^{2} sin (\frac{πn x}{2}) - \frac{16}{π ^{3} n ^{3}} (- \frac{πn}{2} x cos (\frac{πn x}{2}) + sin (\frac{πn x}{2}))) + C

d er i v a t i v e x ln (x)

x \to + 2 lim (\frac{( x ^{2} - 2 x )}{( x ^{2} - 4 x + 4 )})

d er i v a t i v e x^{2} y

\frac{d}{d t} (2 sin (2 t))

d er i v a t i v e \frac{3 x + 1}{x ^{2} + 1}