integral of ((7x+29))/(x^2+8x+15)

解答

\int \frac{( 7 x + 29 )}{x ^{2} + 8 x + 15} d x

4 ln ∣ x + 3 ∣ + 3 ln ∣ x + 5 ∣ + C

求解步骤

\int \frac{7 x + 29}{x ^{2} + 8 x + 15} d x

将

\frac{7 x + 29}{x ^{2} + 8 x + 15}

用部份分式展开:

\frac{4}{x + 3} + \frac{3}{x + 5}

= \int \frac{4}{x + 3} + \frac{3}{x + 5} 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

= \int \frac{4}{x + 3} d x + \int \frac{3}{x + 5} d x

\int \frac{4}{x + 3} d x = 4 ln ∣ x + 3 ∣

\int \frac{3}{x + 5} d x = 3 ln ∣ x + 5 ∣

= 4 ln ∣ x + 3 ∣ + 3 ln ∣ x + 5 ∣

解答补常数

= 4 ln ∣ x + 3 ∣ + 3 ln ∣ x + 5 ∣ + C

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

d er i v a t i v e 1 + x^{3}

x \to 0 lim (2 x - \frac{1}{x ^{2}})

in v erse l a pl a ce \frac{1}{( s - 2 ) ( s + 1 ) ( s - 3 )}

d er i v a t i v e 6 e^{6 e^{x} + x}