integral of 10(x+1)(x^2+2x+3)^4

解答

\int 10 (x + 1) (x^{2} + 2 x + 3)^{4} d x

x^{10} + 10 x^{9} + 55 x^{8} + 200 x^{7} + 530 x^{6} + 1052 x^{5} + 1590 x^{4} + 1800 x^{3} + 1485 x^{2} + 810 x + C

求解步骤

\int 10 (x + 1) (x^{2} + 2 x + 3)^{4} d x

提出常数:

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

= 10 \cdot \int (x + 1) (x^{2} + 2 x + 3)^{4} d x

乘开

(x + 1) (x^{2} + 2 x + 3)^{4} : x^{9} + 9 x^{8} + 44 x^{7} + 140 x^{6} + 318 x^{5} + 526 x^{4} + 636 x^{3} + 540 x^{2} + 297 x + 81

= 10 \cdot \int x^{9} + 9 x^{8} + 44 x^{7} + 140 x^{6} + 318 x^{5} + 526 x^{4} + 636 x^{3} + 540 x^{2} + 297 x + 81 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

= 10 (\int x^{9} d x + \int 9 x^{8} d x + \int 44 x^{7} d x + \int 140 x^{6} d x + \int 318 x^{5} d x + \int 526 x^{4} d x + \int 636 x^{3} d x + \int 540 x^{2} d x + \int 297 x d x + \int 81 d x)

\int x^{9} d x = \frac{x ^{10}}{10}

\int 9 x^{8} d x = x^{9}

\int 44 x^{7} d x = \frac{11 x ^{8}}{2}

\int 140 x^{6} d x = 20 x^{7}

\int 318 x^{5} d x = 53 x^{6}

\int 526 x^{4} d x = \frac{526 x ^{5}}{5}

\int 636 x^{3} d x = 159 x^{4}

\int 540 x^{2} d x = 180 x^{3}

\int 297 x d x = \frac{297 x ^{2}}{2}

\int 81 d x = 81 x

= 10 (\frac{x ^{10}}{10} + x^{9} + \frac{11 x ^{8}}{2} + 20 x^{7} + 53 x^{6} + \frac{526 x ^{5}}{5} + 159 x^{4} + 180 x^{3} + \frac{297 x ^{2}}{2} + 81 x)

化简

10 (\frac{x ^{10}}{10} + x^{9} + \frac{11 x ^{8}}{2} + 20 x^{7} + 53 x^{6} + \frac{526 x ^{5}}{5} + 159 x^{4} + 180 x^{3} + \frac{297 x ^{2}}{2} + 81 x) : x^{10} + 10 x^{9} + 55 x^{8} + 200 x^{7} + 530 x^{6} + 1052 x^{5} + 1590 x^{4} + 1800 x^{3} + 1485 x^{2} + 810 x

= x^{10} + 10 x^{9} + 55 x^{8} + 200 x^{7} + 530 x^{6} + 1052 x^{5} + 1590 x^{4} + 1800 x^{3} + 1485 x^{2} + 810 x

解答补常数

= x^{10} + 10 x^{9} + 55 x^{8} + 200 x^{7} + 530 x^{6} + 1052 x^{5} + 1590 x^{4} + 1800 x^{3} + 1485 x^{2} + 810 x + C

\frac{d}{d x} (a r (x) cos (\frac{x + 1}{3}))

\frac{d}{d x} (18 x - \frac{2}{3} x^{3})

x \to 1 lim (e^{x} - e)

\int_{3}^{\infty} x e^{- x} d x

\int \frac{1}{x ^{4} x ^{2} - 2} d x