zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of (9x(x^2-10)^8)/5

解答

\int \frac{9 x ( x ^{2} - 10 ) ^{8}}{5} d x

解答

\frac{9}{10} (x^{2} (x^{2} - 10)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 10)^{7} - \frac{1}{36} (14 x^{18} - 945 x^{16} + 27000 x^{14} - 420000 x^{12} + 3780000 x^{10} - 18900000 x^{8} + 42000000 x^{6}))) + C

求解步骤

\int \frac{9 x ( x ^{2} - 10 ) ^{8}}{5} d x

提出常数:

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

= \frac{9}{5} \cdot \int x (x^{2} - 10)^{8} d x

使用换元积分法

= \frac{9}{5} \cdot \int \frac{( u - 10 ) ^{8}}{2} d u

提出常数:

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

= \frac{9}{5} \cdot \frac{1}{2} \cdot \int (u - 10)^{8} d u

使用分布积分法

= \frac{9}{5} \cdot \frac{1}{2} ((u - 10)^{8} u - \int 8 (u - 10)^{7} u d u)

\int 8 (u - 10)^{7} u d u = 8 (\frac{1}{2} u^{2} (u - 10)^{7} - \frac{1}{36} (14 u^{9} - 945 u^{8} + 27000 u^{7} - 420000 u^{6} + 3780000 u^{5} - 18900000 u^{4} + 42000000 u^{3}))

= \frac{9}{5} \cdot \frac{1}{2} ((u - 10)^{8} u - 8 (\frac{1}{2} u^{2} (u - 10)^{7} - \frac{1}{36} (14 u^{9} - 945 u^{8} + 27000 u^{7} - 420000 u^{6} + 3780000 u^{5} - 18900000 u^{4} + 42000000 u^{3})))

u = x^{2}

代回

= \frac{9}{5} \cdot \frac{1}{2} ((x^{2} - 10)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 10)^{7} - \frac{1}{36} (14 (x^{2})^{9} - 945 (x^{2})^{8} + 27000 (x^{2})^{7} - 420000 (x^{2})^{6} + 3780000 (x^{2})^{5} - 18900000 (x^{2})^{4} + 42000000 (x^{2})^{3})))

化简

\frac{9}{5} \cdot \frac{1}{2} ((x^{2} - 10)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 10)^{7} - \frac{1}{36} (14 (x^{2})^{9} - 945 (x^{2})^{8} + 27000 (x^{2})^{7} - 420000 (x^{2})^{6} + 3780000 (x^{2})^{5} - 18900000 (x^{2})^{4} + 42000000 (x^{2})^{3}))) : \frac{9}{10} (x^{2} (x^{2} - 10)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 10)^{7} - \frac{1}{36} (14 x^{18} - 945 x^{16} + 27000 x^{14} - 420000 x^{12} + 3780000 x^{10} - 18900000 x^{8} + 42000000 x^{6})))

= \frac{9}{10} (x^{2} (x^{2} - 10)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 10)^{7} - \frac{1}{36} (14 x^{18} - 945 x^{16} + 27000 x^{14} - 420000 x^{12} + 3780000 x^{10} - 18900000 x^{8} + 42000000 x^{6})))

解答补常数

= \frac{9}{10} (x^{2} (x^{2} - 10)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 10)^{7} - \frac{1}{36} (14 x^{18} - 945 x^{16} + 27000 x^{14} - 420000 x^{12} + 3780000 x^{10} - 18900000 x^{8} + 42000000 x^{6}))) + C

作图

流行的例子

integral of x/(sin^2(x^2+1))

integral of 1/(sqrt(-x^2+6x))

integral of (sin(2x))/(4+cos(2x))

integral of (sin^5(x))/(cos^7(x))

integral of (2t-1)/(t^2+t)