integral from 0 to 3 of (2x+3)/(1+x^2)

解答

\int_{0}^{3} \frac{2 x + 3}{1 + x ^{2}} d x

ln (10) + 3 arctan (3)

十进制

6.04972 \dots

求解步骤

\int_{0}^{3} \frac{2 x + 3}{1 + x ^{2}} d x

乘开

\frac{2 x + 3}{1 + x ^{2}} : \frac{2 x}{1 + x ^{2}} + \frac{3}{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_{0}^{3} \frac{2 x}{1 + x ^{2}} d x + \int_{0}^{3} \frac{3}{1 + x ^{2}} d x

\int_{0}^{3} \frac{2 x}{1 + x ^{2}} d x = ln (10)

\int_{0}^{3} \frac{3}{1 + x ^{2}} d x = 3 arctan (3)

= ln (10) + 3 arctan (3)

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