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受欢迎的微积分 >

integral of 1/(x(ln(x^3)))

解答

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

解答

\frac{1}{3} ln (3 ∣ ln (x) ∣) + C

求解步骤

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

使用换元积分法

= \int \frac{1}{3 u ln ( u )} d u

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}{u ln ( u )} d u

使用换元积分法

= \frac{1}{3} \cdot \int \frac{1}{v} d v

使用积分常见定则:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{3} ln ∣ v ∣

代回

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

化简

\frac{1}{3} ln ln (x^{3}) : \frac{1}{3} ln (3 ∣ ln (x) ∣)

= \frac{1}{3} ln (3 ∣ ln (x) ∣)

解答补常数

= \frac{1}{3} ln (3 ∣ ln (x) ∣) + C

作图

流行的例子

sum from n=1 to infinity of 5/(n^2+64)

n = 1 \sum \infty \frac{5}{n ^{2} + 64}

integral from 1 to 6x of 7/t

\int_{1}^{6 x} \frac{7}{t} d t

integral of cot^5(x)

\int cot^{5} (x) d x

derivative of (x+4/x)

\frac{d}{d x} (\frac{x + 4}{x})

derivative of (x^3+ix^2+x+1/(x^2+i))

\frac{d}{d x} (\frac{x ^{3} + i x ^{2} + x + 1}{x ^{2} + i})