integral of 3/(x^3+x)

解答

\int \frac{3}{x ^{3} + x} d x

3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

求解步骤

\int \frac{3}{x ^{3} + x} d x

提出常数:

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

= 3 \cdot \int \frac{1}{x ^{3} + x} d x

将

\frac{1}{x ^{3} + x}

用部份分式展开:

\frac{1}{x} - \frac{x}{x ^{2} + 1}

= 3 \cdot \int \frac{1}{x} - \frac{x}{x ^{2} + 1} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (\int \frac{1}{x} d x - \int \frac{x}{x ^{2} + 1} d x)

\int \frac{1}{x} d x = ln ∣ x ∣

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

= 3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

解答补常数

= 3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

\int_{0}^{2} t e^{2 t} d t

\frac{\partial}{\partial x} (x^{2} e^{3 x y})

\int (\frac{x}{x x ^{2} + 36}) d x

\frac{\partial}{\partial y} (ln (x + y^{2}))

\int_{0}^{5} \frac{1}{x ^{\frac{20}{21}}} d x