integral of (22x^2)/(x^4-61x^2+900)

解答

\int \frac{22 x ^{2}}{x ^{4} - 61 x ^{2} + 900} d x

5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 6 ln ∣ x + 6 ∣ + 6 ln ∣ x - 6 ∣ + C

求解步骤

\int \frac{22 x ^{2}}{x ^{4} - 61 x ^{2} + 900} d x

提出常数:

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

= 22 \cdot \int \frac{x ^{2}}{x ^{4} - 61 x ^{2} + 900} d x

将

\frac{x ^{2}}{x ^{4} - 61 x ^{2} + 900}

用部份分式展开:

\frac{5}{22 ( x + 5 )} - \frac{5}{22 ( x - 5 )} - \frac{3}{11 ( x + 6 )} + \frac{3}{11 ( x - 6 )}

= 22 \cdot \int \frac{5}{22 ( x + 5 )} - \frac{5}{22 ( x - 5 )} - \frac{3}{11 ( x + 6 )} + \frac{3}{11 ( x - 6 )} 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

= 22 (\int \frac{5}{22 ( x + 5 )} d x - \int \frac{5}{22 ( x - 5 )} d x - \int \frac{3}{11 ( x + 6 )} d x + \int \frac{3}{11 ( x - 6 )} d x)

\int \frac{5}{22 ( x + 5 )} d x = \frac{5}{22} ln ∣ x + 5 ∣

\int \frac{5}{22 ( x - 5 )} d x = \frac{5}{22} ln ∣ x - 5 ∣

\int \frac{3}{11 ( x + 6 )} d x = \frac{3}{11} ln ∣ x + 6 ∣

\int \frac{3}{11 ( x - 6 )} d x = \frac{3}{11} ln ∣ x - 6 ∣

= 22 (\frac{5}{22} ln ∣ x + 5 ∣ - \frac{5}{22} ln ∣ x - 5 ∣ - \frac{3}{11} ln ∣ x + 6 ∣ + \frac{3}{11} ln ∣ x - 6 ∣)

化简

22 (\frac{5}{22} ln ∣ x + 5 ∣ - \frac{5}{22} ln ∣ x - 5 ∣ - \frac{3}{11} ln ∣ x + 6 ∣ + \frac{3}{11} ln ∣ x - 6 ∣) : 5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 6 ln ∣ x + 6 ∣ + 6 ln ∣ x - 6 ∣

= 5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 6 ln ∣ x + 6 ∣ + 6 ln ∣ x - 6 ∣

解答补常数

= 5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 6 ln ∣ x + 6 ∣ + 6 ln ∣ x - 6 ∣ + C

x \to 7 lim (\frac{7 - x}{7 - x})

x \to 6 lim (- 110)

\int \frac{x ^{4}}{x ^{5} - 4} d x

\frac{d}{d x} (\frac{4}{3} π \cdot x^{3})

\int_{- 1}^{9} (9 - x^{2} + 8 x) d x