integral of (x^7e^x-4x^6)/(x^7)

解答

\int \frac{x ^{7} e ^{x} - 4 x ^{6}}{x ^{7}} d x

e^{x} - 4 ln ∣ x ∣ + C

求解步骤

\int \frac{x ^{7} e ^{x} - 4 x ^{6}}{x ^{7}} d x

乘开

\frac{x ^{7} e ^{x} - 4 x ^{6}}{x ^{7}} : e^{x} - \frac{4}{x}

= \int e^{x} - \frac{4}{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 e^{x} d x - \int \frac{4}{x} d x

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

= e^{x} - 4 ln ∣ x ∣

解答补常数

= e^{x} - 4 ln ∣ x ∣ + C

\int \frac{ln ( x )}{x ^{7}} d x

y^{^{''}} - y^{^{'}} - 2 y = - 8 t + 6 t^{2}

x \to 2 - lim (\frac{3 x + 5}{x - 2})

\int \frac{2 t}{1 + t ^{2}} d t

a re a y = - 2 x^{2} + 20, y = 3 x - 15