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受欢迎的微积分 >

integral of (7x)/(x^2+14x+40)

解答

\int \frac{7 x}{x ^{2} + 14 x + 40} d x

解答

7 (\frac{5}{3} ln ∣ x + 10 ∣ - \frac{2}{3} ln ∣ x + 4 ∣) + C

求解步骤

\int \frac{7 x}{x ^{2} + 14 x + 40} d x

提出常数:

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

= 7 \cdot \int \frac{x}{x ^{2} + 14 x + 40} d x

对

x^{2} + 14 x + 40

配方:

(x + 7)^{2} - 9

= 7 \cdot \int \frac{x}{( x + 7 ) ^{2} - 9} d x

使用换元积分法

= 7 \cdot \int \frac{u - 7}{u ^{2} - 9} d u

将

\frac{u - 7}{u ^{2} - 9}

用部份分式展开:

\frac{5}{3 ( u + 3 )} - \frac{2}{3 ( u - 3 )}

= 7 \cdot \int \frac{5}{3 ( u + 3 )} - \frac{2}{3 ( u - 3 )} d u

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

= 7 (\int \frac{5}{3 ( u + 3 )} d u - \int \frac{2}{3 ( u - 3 )} d u)

\int \frac{5}{3 ( u + 3 )} d u = \frac{5}{3} ln ∣ u + 3 ∣

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

= 7 (\frac{5}{3} ln ∣ u + 3 ∣ - \frac{2}{3} ln ∣ u - 3 ∣)

u = x + 7

代回

= 7 (\frac{5}{3} ln ∣ x + 7 + 3 ∣ - \frac{2}{3} ln ∣ x + 7 - 3 ∣)

化简

7 (\frac{5}{3} ln ∣ x + 7 + 3 ∣ - \frac{2}{3} ln ∣ x + 7 - 3 ∣) : 7 (\frac{5}{3} ln ∣ x + 10 ∣ - \frac{2}{3} ln ∣ x + 4 ∣)

= 7 (\frac{5}{3} ln ∣ x + 10 ∣ - \frac{2}{3} ln ∣ x + 4 ∣)

解答补常数

= 7 (\frac{5}{3} ln ∣ x + 10 ∣ - \frac{2}{3} ln ∣ x + 4 ∣) + C

作图

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