integral of (4+5x)/(1+x^2)

解答

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

4 arctan (x) + \frac{5}{2} ln 1 + x^{2} + C

求解步骤

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

乘开

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

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

\int \frac{5 x}{1 + x ^{2}} d x = \frac{5}{2} ln 1 + x^{2}

= 4 arctan (x) + \frac{5}{2} ln 1 + x^{2}

解答补常数

= 4 arctan (x) + \frac{5}{2} ln 1 + x^{2} + C

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