zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of (6x^2)/(x^4-5x^2+4)

解答

\int \frac{6 x ^{2}}{x ^{4} - 5 x ^{2} + 4} d x

解答

ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣ + C

求解步骤

\int \frac{6 x ^{2}}{x ^{4} - 5 x ^{2} + 4} d x

提出常数:

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

= 6 \cdot \int \frac{x ^{2}}{x ^{4} - 5 x ^{2} + 4} d x

将

\frac{x ^{2}}{x ^{4} - 5 x ^{2} + 4}

用部份分式展开:

\frac{1}{6 ( x + 1 )} - \frac{1}{6 ( x - 1 )} - \frac{1}{3 ( x + 2 )} + \frac{1}{3 ( x - 2 )}

= 6 \cdot \int \frac{1}{6 ( x + 1 )} - \frac{1}{6 ( x - 1 )} - \frac{1}{3 ( x + 2 )} + \frac{1}{3 ( x - 2 )} 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

= 6 (\int \frac{1}{6 ( x + 1 )} d x - \int \frac{1}{6 ( x - 1 )} d x - \int \frac{1}{3 ( x + 2 )} d x + \int \frac{1}{3 ( x - 2 )} d x)

\int \frac{1}{6 ( x + 1 )} d x = \frac{1}{6} ln ∣ x + 1 ∣

\int \frac{1}{6 ( x - 1 )} d x = \frac{1}{6} ln ∣ x - 1 ∣

\int \frac{1}{3 ( x + 2 )} d x = \frac{1}{3} ln ∣ x + 2 ∣

\int \frac{1}{3 ( x - 2 )} d x = \frac{1}{3} ln ∣ x - 2 ∣

= 6 (\frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{3} ln ∣ x + 2 ∣ + \frac{1}{3} ln ∣ x - 2 ∣)

化简

6 (\frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{3} ln ∣ x + 2 ∣ + \frac{1}{3} ln ∣ x - 2 ∣) : ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣

= ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣

解答补常数

= ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣ + C

作图

流行的例子

integral of 22tan^6(x)

\int 22 tan^{6} (x) d x

integral of 1/(sqrt(4s^2+5))

\int \frac{1}{4 s ^{2} + 5} d s

integral of 7/4

\int \frac{7}{4} d x

integral of sqrt(x^3-2x^2+x)

\int x^{3} - 2 x^{2} + x d x

integral of sin((npix)/(40))

\int sin (\frac{nπ x}{40}) d x