integral of (1+sin(x))/(x^3)

解答

\int \frac{1 + sin ( x )}{x ^{3}} d x

- \frac{1}{2 x ^{2}} - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x)) + C

求解步骤

\int \frac{1 + sin ( x )}{x ^{3}} d x

乘开

\frac{1 + sin ( x )}{x ^{3}} : \frac{1}{x ^{3}} + \frac{sin ( x )}{x ^{3}}

= \int \frac{1}{x ^{3}} + \frac{sin ( x )}{x ^{3}} 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 \frac{1}{x ^{3}} d x + \int \frac{sin ( x )}{x ^{3}} d x

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

\int \frac{sin ( x )}{x ^{3}} d x = - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x))

= - \frac{1}{2 x ^{2}} - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x))

解答补常数

= - \frac{1}{2 x ^{2}} - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x)) + C

\int_{0}^{9} π (x - 1)^{2} d x

t an g e n t y = 7 e^{x} cos (x), (0, 7)

a re a 4 t^{3} - t^{2}, t^{4} + 2 t^{2}, 1 \leq t \leq 3

\int_{0}^{\infty} t e^{- 3 t} d t

x \to 6 lim (8 (x - 5) (x - 7))