integral of e^xsin(x)

解答

\int e^{x} sin (x) d x

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

求解步骤

\int e^{x} sin (x) d x

使用分布积分法

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

提出常数:

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

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

使用分布积分法

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

因此

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

整理

\int e^{x} sin (x) d x

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

解答补常数

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

d er i v a t i v e f (x) = \frac{1}{27} (9 x^{2} + 6)^{\frac{3}{2}}

\int e^{\frac{1}{2} x^{2}} d x

x \to 4 - lim (∣ x - 4 ∣)

y^{^{'}} = y + x^{2} e^{x} + 1 - \frac{1}{e}

\frac{\partial}{\partial x} (3 x y^{2} + 2 y cos (x))