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 1/((u^2+a^2)^2)

解答

\int \frac{1}{( u ^{2} + a ^{2} ) ^{2}} d u

解答

\frac{1}{2 a ^{3}} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a}))) + C

求解步骤

\int \frac{1}{( u ^{2} + a ^{2} ) ^{2}} d u

a^{2} + u^{2} = a^{2} (1 + \frac{u ^{2}}{a ^{2}})

= \int \frac{1}{( a ^{2} ( 1 + \frac{u ^{2}}{a ^{2}} ) ) ^{2}} d u

使用换元积分法

= \int \frac{1}{a ^{3} ( v ^{2} + 1 ) ^{2}} d v

提出常数:

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

= \frac{1}{a ^{3}} \cdot \int \frac{1}{( v ^{2} + 1 ) ^{2}} d v

使用三角换元法

= \frac{1}{a ^{3}} \cdot \int \frac{1}{sec ^{4} ( w ) cos ^{2} ( w )} d w

使用三角恒等式改写

= \frac{1}{a ^{3}} \cdot \int \frac{1 + cos ( 2 w )}{2} d w

提出常数:

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

= \frac{1}{a ^{3}} \cdot \frac{1}{2} \cdot \int 1 + cos (2 w) d w

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

= \frac{1}{a ^{3}} \cdot \frac{1}{2} (\int 1 d w + \int cos (2 w) d w)

\int 1 d w = w

\int cos (2 w) d w = \frac{1}{2} sin (2 w)

= \frac{1}{a ^{3}} \cdot \frac{1}{2} (w + \frac{1}{2} sin (2 w))

代回

= \frac{1}{a ^{3}} \cdot \frac{1}{2} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a})))

化简

\frac{1}{a ^{3}} \cdot \frac{1}{2} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a}))) : \frac{1}{2 a ^{3}} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a})))

= \frac{1}{2 a ^{3}} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a})))

解答补常数

= \frac{1}{2 a ^{3}} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a}))) + C

作图

流行的例子

y^{''}+10y^'+26y=0

y^{^{''}} + 10 y^{^{'}} + 26 y = 0

derivative f(x)=x^2sqrt(9-x^2)

d er i v a t i v e f (x) = x^{2} 9 - x^{2}

x^4+y^3sqrt(x^5+3)y^'=0

x^{4} + y^{3} x^{5} + 3 y^{^{'}} = 0

derivative of x^2(x+7^3)

\frac{d}{d x} (x^{2} (x + 7)^{3})

x(dy)/(dx)+2y=10x^2

x \frac{d y}{d x} + 2 y = 10 x^{2}