integral of 1/(x^3sqrt(ln(x^5)))

解答

\int \frac{1}{x ^{3} ln ( x ^{5} )} d x

\frac{π}{10} erf (2 ln (x)) + C

求解步骤

\int \frac{1}{x ^{3} ln ( x ^{5} )} d x

ln (x^{5}) = 5 ln (x)

= \int \frac{1}{5 x ^{3} ln ( x )} d x

提出常数:

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

= \frac{1}{5} \cdot \int \frac{1}{x ^{3} ln ( x )} d x

使用换元积分法

= \frac{1}{5} \cdot \int \frac{2}{e ^{2 u^{2}}} d u

提出常数:

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

= \frac{1}{5} \cdot 2 \cdot \int \frac{1}{e ^{2 u^{2}}} d u

使用分布积分法

= \frac{1}{5} \cdot 2 (\frac{u}{e ^{2 u^{2}}} - \int - 4 e^{- 2 u^{2}} u^{2} d u)

\int - 4 e^{- 2 u^{2}} u^{2} d u = e^{- 2 u^{2}} u - \frac{π}{2 2} erf (2 u)

= \frac{1}{5} \cdot 2 (\frac{u}{e ^{2 u^{2}}} - (e^{- 2 u^{2}} u - \frac{π}{2 2} erf (2 u)))

u = ln (x)

代回

= \frac{1}{5} \cdot 2 (\frac{ln ( x )}{e ^{2 (l n (x))^{2}}} - (e^{- 2 (l n (x))^{2}} ln (x) - \frac{π}{2 2} erf (2 ln (x))))

化简

\frac{1}{5} \cdot 2 (\frac{ln ( x )}{e ^{2 (l n (x))^{2}}} - (e^{- 2 (l n (x))^{2}} ln (x) - \frac{π}{2 2} erf (2 ln (x)))) : \frac{π}{10} erf (2 ln (x))

= \frac{π}{10} erf (2 ln (x))

解答补常数

= \frac{π}{10} erf (2 ln (x)) + C

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