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 (sqrt(x^2-36))/(x^4)

解答

\int \frac{x ^{2} - 36}{x ^{4}} d x

解答

\frac{( x ^{2} - 36 ) ^{\frac{3}{2}}}{108 x ^{3}} + C

求解步骤

\int \frac{x ^{2} - 36}{x ^{4}} d x

使用三角换元法

= \int \frac{tan ^{2} ( u )}{36 sec ^{3} ( u )} d u

提出常数:

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

= \frac{1}{36} \cdot \int \frac{tan ^{2} ( u )}{sec ^{3} ( u )} d u

使用三角恒等式改写

= \frac{1}{36} \cdot \int sin^{2} (u) cos (u) d u

使用换元积分法

= \frac{1}{36} \cdot \int v^{2} d v

使用幂法则

= \frac{1}{36} \cdot \frac{v ^{3}}{3}

代回

= \frac{1}{36} \cdot \frac{sin ^{3} ( arcsec ( \frac{1}{6} x ) )}{3}

化简

\frac{1}{36} \cdot \frac{sin ^{3} ( arcsec ( \frac{1}{6} x ) )}{3} : \frac{( x ^{2} - 36 ) ^{\frac{3}{2}}}{108 x ^{3}}

= \frac{( x ^{2} - 36 ) ^{\frac{3}{2}}}{108 x ^{3}}

解答补常数

= \frac{( x ^{2} - 36 ) ^{\frac{3}{2}}}{108 x ^{3}} + C

作图

流行的例子

integral from 1 to 2 of (-2x+6)

\int_{1}^{2} (- 2 x + 6) d x

t an g e n t 3 x^{2} + 2

derivative y=(x^2+3x)2

d er i v a t i v e y = (x^{2} + 3 x) 2

(\partial)/(\partial x)(5x^2+y^2+9)

\frac{\partial}{\partial x} (5 x^{2} + y^{2} + 9)

integral of (cos^2(θ))^2

\int (cos^{2} (θ))^{2} d θ