integral of ((2x-1))/((x-3)^2)

解答

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

2 ln ∣ x - 3 ∣ - \frac{5}{x - 3} + C

求解步骤

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

乘开

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

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

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

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

= 2 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - (- \frac{1}{x - 3})

化简

2 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - (- \frac{1}{x - 3}) : 2 ln ∣ x - 3 ∣ - \frac{5}{x - 3}

= 2 ln ∣ x - 3 ∣ - \frac{5}{x - 3}

解答补常数

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

\int 9 e^{6 y} d y

\int \frac{1}{( 8 x + 6 ) ^{4}} d x

\int 3 sin (x) + 2 x d x

\int \frac{4}{x ^{5}} cos (\frac{1}{x ^{2}}) d x

\int 2 csc^{2} (x) - 3 sec^{2} (x) d x