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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

提出常数:

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

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

化简

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

使用换元积分法

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

使用三角换元法

= 4 \cdot \int sec (v) d v

使用积分常见定则:

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

= 4 ln ∣ tan (v) + sec (v) ∣

代回

= 4 ln tan (arctan (\frac{1}{3} ln (x))) + sec (arctan (\frac{1}{3} ln (x)))

化简

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

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

解答补常数

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

作图

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