integral of 1/(x(9+ln^2(x)))

解答

\int \frac{1}{x ( 9 + ln ^{2} ( x ) )} d x

\frac{1}{3} arctan (\frac{ln ( x )}{3}) + C

求解步骤

\int \frac{1}{x ( 9 + ln ^{2} ( x ) )} d x

使用换元积分法

= \int \frac{1}{9 + u ^{2}} d u

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

= \frac{1}{3} arctan (v)

代回

= \frac{1}{3} arctan (\frac{ln ( x )}{3})

解答补常数

= \frac{1}{3} arctan (\frac{ln ( x )}{3}) + C

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