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受欢迎的微积分 >

integral of 1/(x(4+(3ln(x))^2))

解答

\int \frac{1}{x ( 4 + ( 3 ln ( x ) ) ^{2} )} d x

解答

\frac{1}{6} arctan (\frac{3}{2} ln (x)) + C

求解步骤

\int \frac{1}{x ( 4 + ( 3 ln ( x ) ) ^{2} )} d x

化简

\frac{1}{x ( 4 + ( 3 ln ( x ) ) ^{2} )} : \frac{1}{x ( 4 + 9 ln ^{2} ( x ) )}

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

使用换元积分法

= \int \frac{1}{4 + 9 u ^{2}} d u

使用换元积分法

= \int \frac{1}{6 ( 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}{6} \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}{6} arctan (v)

代回

= \frac{1}{6} arctan (\frac{3}{2} ln (x))

解答补常数

= \frac{1}{6} arctan (\frac{3}{2} ln (x)) + C

作图

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