integral of ((2x-1))/(((x+1)(x^2+4x+4)))

解答

\int \frac{( 2 x - 1 )}{( ( x + 1 ) ( x ^{2} + 4 x + 4 ) )} d x

- 3 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣ - \frac{5}{x + 2} + C

求解步骤

\int \frac{2 x - 1}{( x + 1 ) ( x ^{2} + 4 x + 4 )} d x

将

\frac{2 x - 1}{( x + 1 ) ( x ^{2} + 4 x + 4 )}

用部份分式展开:

- \frac{3}{x + 1} + \frac{3}{x + 2} + \frac{5}{( x + 2 ) ^{2}}

= \int - \frac{3}{x + 1} + \frac{3}{x + 2} + \frac{5}{( x + 2 ) ^{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

= - \int \frac{3}{x + 1} d x + \int \frac{3}{x + 2} d x + \int \frac{5}{( x + 2 ) ^{2}} d x

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

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

\int \frac{5}{( x + 2 ) ^{2}} d x = - \frac{5}{x + 2}

= - 3 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣ - \frac{5}{x + 2}

解答补常数

= - 3 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣ - \frac{5}{x + 2} + C

\int - 3 e^{\frac{- x}{3}} d x

\int \frac{1}{1 - 2 y} d y

\int cos^{4} (4 x) \cdot sin^{2} (4 x) d x

\int \frac{1}{3 y ^{2} + 1} d y

\int \frac{x + 1}{x ^{3} + 5 x ^{2} + 6 x} d x