integral of (90x^2)/(x^4-53x^2+196)

解答

\int \frac{90 x ^{2}}{x ^{4} - 53 x ^{2} + 196} d x

2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 2 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣ + C

求解步骤

\int \frac{90 x ^{2}}{x ^{4} - 53 x ^{2} + 196} d x

提出常数:

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

= 90 \cdot \int \frac{x ^{2}}{x ^{4} - 53 x ^{2} + 196} d x

将

\frac{x ^{2}}{x ^{4} - 53 x ^{2} + 196}

用部份分式展开:

\frac{1}{45 ( x + 2 )} - \frac{1}{45 ( x - 2 )} - \frac{7}{90 ( x + 7 )} + \frac{7}{90 ( x - 7 )}

= 90 \cdot \int \frac{1}{45 ( x + 2 )} - \frac{1}{45 ( x - 2 )} - \frac{7}{90 ( x + 7 )} + \frac{7}{90 ( x - 7 )} 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

= 90 (\int \frac{1}{45 ( x + 2 )} d x - \int \frac{1}{45 ( x - 2 )} d x - \int \frac{7}{90 ( x + 7 )} d x + \int \frac{7}{90 ( x - 7 )} d x)

\int \frac{1}{45 ( x + 2 )} d x = \frac{1}{45} ln ∣ x + 2 ∣

\int \frac{1}{45 ( x - 2 )} d x = \frac{1}{45} ln ∣ x - 2 ∣

\int \frac{7}{90 ( x + 7 )} d x = \frac{7}{90} ln ∣ x + 7 ∣

\int \frac{7}{90 ( x - 7 )} d x = \frac{7}{90} ln ∣ x - 7 ∣

= 90 (\frac{1}{45} ln ∣ x + 2 ∣ - \frac{1}{45} ln ∣ x - 2 ∣ - \frac{7}{90} ln ∣ x + 7 ∣ + \frac{7}{90} ln ∣ x - 7 ∣)

化简

90 (\frac{1}{45} ln ∣ x + 2 ∣ - \frac{1}{45} ln ∣ x - 2 ∣ - \frac{7}{90} ln ∣ x + 7 ∣ + \frac{7}{90} ln ∣ x - 7 ∣) : 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 2 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣

= 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 2 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣

解答补常数

= 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 2 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣ + C

\int \frac{1}{15 - 3 x} d x

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