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受欢迎的微积分 >

integral of (4ln(4x^4))/(x^4)

解答

\int \frac{4 ln ( 4 x ^{4} )}{x ^{4}} d x

解答

8 (- \frac{ln ( 2 )}{3 x ^{3}} + 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}})) + C

求解步骤

\int \frac{4 ln ( 4 x ^{4} )}{x ^{4}} d x

提出常数:

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

= 4 \cdot \int \frac{ln ( 4 x ^{4} )}{x ^{4}} d x

化简

ln (4 x^{4}) : 2 (2 ln (x) + ln (2))

= 4 \cdot \int \frac{2 ( ln ( 2 ) + 2 ln ( x ) )}{x ^{4}} d x

提出常数:

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

= 4 \cdot 2 \cdot \int \frac{ln ( 2 ) + 2 ln ( x )}{x ^{4}} d x

乘开

\frac{ln ( 2 ) + 2 ln ( x )}{x ^{4}} : \frac{ln ( 2 )}{x ^{4}} + \frac{2 ln ( x )}{x ^{4}}

= 4 \cdot 2 \cdot \int \frac{ln ( 2 )}{x ^{4}} + \frac{2 ln ( x )}{x ^{4}} 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

= 4 \cdot 2 (\int \frac{ln ( 2 )}{x ^{4}} d x + \int \frac{2 ln ( x )}{x ^{4}} d x)

\int \frac{ln ( 2 )}{x ^{4}} d x = - \frac{ln ( 2 )}{3 x ^{3}}

\int \frac{2 ln ( x )}{x ^{4}} d x = 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}})

= 4 \cdot 2 (- \frac{ln ( 2 )}{3 x ^{3}} + 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}}))

化简

= 8 (- \frac{ln ( 2 )}{3 x ^{3}} + 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}}))

解答补常数

= 8 (- \frac{ln ( 2 )}{3 x ^{3}} + 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}})) + C

作图

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