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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{2} arctan (\frac{ln ( x )}{2}) + C

求解步骤

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

乘开

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

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

使用换元积分法

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

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

= \frac{1}{2} arctan (v)

代回

= \frac{1}{2} arctan (\frac{ln ( x )}{2})

解答补常数

= \frac{1}{2} arctan (\frac{ln ( x )}{2}) + C

作图

流行的例子

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\int \frac{1}{16 - x ^{4}} d x

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