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

解答

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

\frac{1}{4} ln x^{2} + 4 + \frac{1}{2} ln ∣ x ∣ + C

求解步骤

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

化简

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

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

乘开

\frac{x ^{2} + 2}{x ( x ^{2} + 4 )} : \frac{x ^{2}}{x ( x ^{2} + 4 )} + \frac{2}{x ( x ^{2} + 4 )}

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

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

\int \frac{2}{x ( x ^{2} + 4 )} d x = 2 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

= \frac{1}{2} ln x^{2} + 4 + 2 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

化简

\frac{1}{2} ln x^{2} + 4 + 2 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4) : \frac{1}{4} ln x^{2} + 4 + \frac{1}{2} ln ∣ x ∣

= \frac{1}{4} ln x^{2} + 4 + \frac{1}{2} ln ∣ x ∣

解答补常数

= \frac{1}{4} ln x^{2} + 4 + \frac{1}{2} ln ∣ x ∣ + C

\int (2 x - 3) (4 x^{2} + 6 x + 9) d x

\int \frac{x ^{3}}{x ^{4} - 4} d x

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

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

\int \frac{1}{tan ^{4} ( x )} d x