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受欢迎的微积分 >

integral of (4sqrt(ln^2(x)+9))/x

解答

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

解答

36 (\frac{1}{18} ln (x) 9 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{3} ln (x) + 9 + ln^{2} (x))) + C

求解步骤

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

使用换元积分法

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

使用三角换元法

= 4 \cdot \int 9 sec^{3} (v) d v

提出常数:

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

= 4 \cdot 9 \cdot \int sec^{3} (v) d v

使用降次积分法

= 4 \cdot 9 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= 4 \cdot 9 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代回

= 4 \cdot 9 (\frac{sec ^{2} ( arctan ( \frac{1}{3} ln ( x ) ) ) sin ( arctan ( \frac{1}{3} ln ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{3} ln (x))) + sec (arctan (\frac{1}{3} ln (x))))

化简

4 \cdot 9 (\frac{sec ^{2} ( arctan ( \frac{1}{3} ln ( x ) ) ) sin ( arctan ( \frac{1}{3} ln ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{3} ln (x))) + sec (arctan (\frac{1}{3} ln (x)))) : 36 (\frac{1}{18} ln (x) 9 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{3} ln (x) + 9 + ln^{2} (x)))

= 36 (\frac{1}{18} ln (x) 9 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{3} ln (x) + 9 + ln^{2} (x)))

解答补常数

= 36 (\frac{1}{18} ln (x) 9 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{3} ln (x) + 9 + ln^{2} (x))) + C

作图

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