integral of (3x^3+4x)/((x^2+5)^2)

解答

\int \frac{3 x ^{3} + 4 x}{( x ^{2} + 5 ) ^{2}} d x

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

求解步骤

\int \frac{3 x ^{3} + 4 x}{( x ^{2} + 5 ) ^{2}} d x

乘开

\frac{3 x ^{3} + 4 x}{( x ^{2} + 5 ) ^{2}} : \frac{3 x ^{3}}{( x ^{2} + 5 ) ^{2}} + \frac{4 x}{( x ^{2} + 5 ) ^{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{3 x ^{3}}{( x ^{2} + 5 ) ^{2}} d x + \int \frac{4 x}{( x ^{2} + 5 ) ^{2}} d x

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

\int \frac{4 x}{( x ^{2} + 5 ) ^{2}} d x = - \frac{2}{x ^{2} + 5}

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

解答补常数

= \frac{3}{2} (ln x^{2} + 5 + \frac{5}{x ^{2} + 5}) - \frac{2}{x ^{2} + 5} + C

\int \frac{3 x ^{4} - 2 x + 5}{x + 1} d x

\frac{d}{d x} (A x)

\frac{d}{d y} (1 - x y cos (π y))

d er i v a t i v e \frac{x ^{2}}{40}

x y^{^{'}} = e^{x} - y