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

解答

\int (x^{2} - 2 x + 1) cos (x) d x

x^{2} sin (x) + 2 x cos (x) - 2 x sin (x) - sin (x) - 2 cos (x) + C

求解步骤

\int (x^{2} - 2 x + 1) cos (x) d x

乘开

(x^{2} - 2 x + 1) cos (x) : x^{2} cos (x) - 2 x cos (x) + cos (x)

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

\int x^{2} cos (x) d x = x^{2} sin (x) - 2 (- x cos (x) + sin (x))

\int 2 x cos (x) d x = 2 (x sin (x) + cos (x))

\int cos (x) d x = sin (x)

= x^{2} sin (x) - 2 (- x cos (x) + sin (x)) - 2 (x sin (x) + cos (x)) + sin (x)

化简

x^{2} sin (x) - 2 (- x cos (x) + sin (x)) - 2 (x sin (x) + cos (x)) + sin (x) : x^{2} sin (x) + 2 x cos (x) - 2 x sin (x) - sin (x) - 2 cos (x)

= x^{2} sin (x) + 2 x cos (x) - 2 x sin (x) - sin (x) - 2 cos (x)

解答补常数

= x^{2} sin (x) + 2 x cos (x) - 2 x sin (x) - sin (x) - 2 cos (x) + C

\int 4 e^{- x} cos (x) d x

\int e^{- t} \cdot sin (t) d t

\int (2 x^{2} + x - 1) d x

\int \frac{cos ( 2 x ) e ^{5 x}}{2} d x

\int 12 x + 7 d x