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受欢迎的微积分 >

积分 (36)/((x^4-4x+3))

解答

积分

\frac{36}{( x ^{4} - 4 x + 3 )}

解答

36 (\frac{1}{18} ln x^{2} + 2 x + 3 + \frac{1}{18 2} arctan (\frac{1}{2} (x + 1)) - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{6 ( x - 1 )}) + C

求解步骤

\int \frac{36}{x ^{4} - 4 x + 3} d x

提出常数:

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

= 36 \cdot \int \frac{1}{x ^{4} - 4 x + 3} d x

将

\frac{1}{x ^{4} - 4 x + 3}

用部份分式展开:

\frac{2 x + 3}{18 ( x ^{2} + 2 x + 3 )} - \frac{1}{9 ( x - 1 )} + \frac{1}{6 ( x - 1 ) ^{2}}

= 36 \cdot \int \frac{2 x + 3}{18 ( x ^{2} + 2 x + 3 )} - \frac{1}{9 ( x - 1 )} + \frac{1}{6 ( x - 1 ) ^{2}} 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

= 36 (\int \frac{2 x + 3}{18 ( x ^{2} + 2 x + 3 )} d x - \int \frac{1}{9 ( x - 1 )} d x + \int \frac{1}{6 ( x - 1 ) ^{2}} d x)

\int \frac{2 x + 3}{18 ( x ^{2} + 2 x + 3 )} d x = \frac{1}{18 2} (2 ln x^{2} + 2 x + 3 + arctan (\frac{x + 1}{2}))

\int \frac{1}{9 ( x - 1 )} d x = \frac{1}{9} ln ∣ x - 1 ∣

\int \frac{1}{6 ( x - 1 ) ^{2}} d x = - \frac{1}{6 ( x - 1 )}

= 36 (\frac{1}{18 2} (2 ln x^{2} + 2 x + 3 + arctan (\frac{x + 1}{2})) - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{6 ( x - 1 )})

化简

36 (\frac{1}{18 2} (2 ln x^{2} + 2 x + 3 + arctan (\frac{x + 1}{2})) - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{6 ( x - 1 )}) : 36 (\frac{1}{18} ln x^{2} + 2 x + 3 + \frac{1}{18 2} arctan (\frac{1}{2} (x + 1)) - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{6 ( x - 1 )})

= 36 (\frac{1}{18} ln x^{2} + 2 x + 3 + \frac{1}{18 2} arctan (\frac{1}{2} (x + 1)) - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{6 ( x - 1 )})

解答补常数

= 36 (\frac{1}{18} ln x^{2} + 2 x + 3 + \frac{1}{18 2} arctan (\frac{1}{2} (x + 1)) - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{6 ( x - 1 )}) + C

作图

流行的例子

积分 1/(((x+1)(x-2)))

in t e g r a l \frac{1}{( ( x + 1 ) ( x - 2 ) )}

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in t e g r a l \frac{x}{( a + x ) ^{\frac{1}{2}}}

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in t e g r a l \frac{cos ^{3} ( x )}{sin ( x )}

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in t e g r a l \frac{( x ^{2} )}{( ( 2 x + x ^{3} ) ^{3} )}

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in t e g r a l v = \frac{1}{3 t + 2}