integral of x^2ycos(x)+2xysin(x)-y^2e^x

解答

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

y x^{2} sin (x) - y^{2} e^{x} + C

求解步骤

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

\int y^{2} e^{x} d x = y^{2} e^{x}

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

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

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

化简

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

= y x^{2} sin (x) - y^{2} e^{x}

解答补常数

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

\int (\frac{1}{( 25 - x ^{2} ) ^{\frac{3}{2}}}) d x

\int \frac{5 + x}{5 - x} d x

\int \frac{e ^{x} + 1}{x} d x

t an g e n t f (x) = \frac{4}{x + 1}, at x = 1

x \to 25 lim (x - 25)