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

解答

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

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

求解步骤

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

乘开

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

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

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

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

解答补常数

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

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n = 1 \sum \infty \frac{1}{n ^{2} + 4 n + 5}

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