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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

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

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

使用换元积分法

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

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}{( u ^{2} + 1 ) ^{2}} d u

使用三角换元法

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

使用三角恒等式改写

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

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 v + \int cos (2 v) d v)

\int 1 d v = v

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

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

代回

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

化简

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

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

解答补常数

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

作图

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