integral from-3 to x^2 of (x-5)(x^2+1)

解答

\int_{- 3}^{x^{2}} (x - 5) (x^{2} + 1) d x

\frac{1}{4} (x^{8} - 81) - \frac{5 x ^{6}}{3} + \frac{1}{2} (x^{4} - 9) - 5 x^{2} - 60

求解步骤

\int_{- 3}^{x^{2}} (x - 5) (x^{2} + 1) d x

乘开

(x - 5) (x^{2} + 1) : x^{3} + x - 5 x^{2} - 5

= \int_{- 3}^{x^{2}} x^{3} + x - 5 x^{2} - 5 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_{- 3}^{x^{2}} x^{3} d x + \int_{- 3}^{x^{2}} x d x - \int_{- 3}^{x^{2}} 5 x^{2} d x - \int_{- 3}^{x^{2}} 5 d x

\int_{- 3}^{x^{2}} x^{3} d x = \frac{x ^{8} - 81}{4}

\int_{- 3}^{x^{2}} x d x = \frac{x ^{4} - 9}{2}

\int_{- 3}^{x^{2}} 5 x^{2} d x = 5 (\frac{x ^{6}}{3} + 9)

\int_{- 3}^{x^{2}} 5 d x = 5 x^{2} + 15

= \frac{x ^{8} - 81}{4} + \frac{x ^{4} - 9}{2} - 5 (\frac{x ^{6}}{3} + 9) - (5 x^{2} + 15)

化简

\frac{x ^{8} - 81}{4} + \frac{x ^{4} - 9}{2} - 5 (\frac{x ^{6}}{3} + 9) - (5 x^{2} + 15) : \frac{1}{4} (x^{8} - 81) - \frac{5 x ^{6}}{3} + \frac{1}{2} (x^{4} - 9) - 5 x^{2} - 60

= \frac{1}{4} (x^{8} - 81) - \frac{5 x ^{6}}{3} + \frac{1}{2} (x^{4} - 9) - 5 x^{2} - 60

\int_{3}^{5} \frac{1}{x ^{2} - 6 x + 9} d x

\int_{2}^{4} (\frac{1}{x ^{2}}) d x

\int_{0}^{2 π} (x sin (x)) d x

\int_{0}^{t a n (x)} \frac{d t}{1 + t ^{2}} d x

\int_{0}^{1} - 4 x d x