integral of (8-e^x)/(e^{5x)}

解答

\int \frac{8 - e ^{x}}{e ^{5 x}} d x

- \frac{8}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}} + C

求解步骤

\int \frac{8 - e ^{x}}{e ^{5 x}} d x

使用换元积分法

= \int \frac{8 - u}{u ^{6}} d u

乘开

\frac{8 - u}{u ^{6}} : \frac{8}{u ^{6}} - \frac{1}{u ^{5}}

= \int \frac{8}{u ^{6}} - \frac{1}{u ^{5}} d u

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 \frac{8}{u ^{6}} d u - \int \frac{1}{u ^{5}} d u

\int \frac{8}{u ^{6}} d u = - \frac{8}{5 u ^{5}}

\int \frac{1}{u ^{5}} d u = - \frac{1}{4 u ^{4}}

= - \frac{8}{5 u ^{5}} - (- \frac{1}{4 u ^{4}})

u = e^{x}

代回

= - \frac{8}{5 ( e ^{x} ) ^{5}} - (- \frac{1}{4 ( e ^{x} ) ^{4}})

化简

- \frac{8}{5 ( e ^{x} ) ^{5}} - (- \frac{1}{4 ( e ^{x} ) ^{4}}) : - \frac{8}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}}

= - \frac{8}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}}

解答补常数

= - \frac{8}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}} + C

d er i v a t i v e 3^{x} tan (ln (x))

\int \frac{x}{x ^{4}} d x

x \to 0 lim (\frac{tan ( x )}{3 x})

y d y = 4 x (y^{2} + 1)^{\frac{1}{2}} d x

y^{^{''}} + 25 y = - 40 sin (5 t)