integral of-(x^2+11x+28)/((x+2)^2)

解答

\int - \frac{x ^{2} + 11 x + 28}{( x + 2 ) ^{2}} d x

- x - 7 ln ∣ x + 2 ∣ + \frac{10}{x + 2} - 2 + C

求解步骤

\int - \frac{x ^{2} + 11 x + 28}{( x + 2 ) ^{2}} d x

提出常数:

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

= - \int \frac{x ^{2} + 11 x + 28}{( x + 2 ) ^{2}} d x

乘开

\frac{x ^{2} + 11 x + 28}{( x + 2 ) ^{2}} : \frac{x ^{2}}{( x + 2 ) ^{2}} + \frac{11 x}{( x + 2 ) ^{2}} + \frac{28}{( x + 2 ) ^{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{x ^{2}}{( x + 2 ) ^{2}} d x + \int \frac{11 x}{( x + 2 ) ^{2}} d x + \int \frac{28}{( x + 2 ) ^{2}} d x)

\int \frac{x ^{2}}{( x + 2 ) ^{2}} d x = x + 2 - 4 ln ∣ x + 2 ∣ - \frac{4}{x + 2}

\int \frac{11 x}{( x + 2 ) ^{2}} d x = 11 (ln ∣ x + 2 ∣ + \frac{2}{x + 2})

\int \frac{28}{( x + 2 ) ^{2}} d x = - \frac{28}{x + 2}

= - (x + 2 - 4 ln ∣ x + 2 ∣ - \frac{4}{x + 2} + 11 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{28}{x + 2})

化简

- (x + 2 - 4 ln ∣ x + 2 ∣ - \frac{4}{x + 2} + 11 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{28}{x + 2}) : - x - 7 ln ∣ x + 2 ∣ + \frac{10}{x + 2} - 2

= - x - 7 ln ∣ x + 2 ∣ + \frac{10}{x + 2} - 2

解答补常数

= - x - 7 ln ∣ x + 2 ∣ + \frac{10}{x + 2} - 2 + C

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

d er i v a t i v e f (x) = - 16 t^{2}

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

t an g e n t 2 x^{2} - 5 x + 3

t a y l or sin (x - x^{2})