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

解答

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

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

求解步骤

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

使用换元积分法

= \int_{4 x^{2} - 4 x + 2}^{2} \frac{v - 1}{2 v} d v

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{4 x^{2} - 4 x + 2}^{2} \frac{v - 1}{v} d v

乘开

\frac{v - 1}{v} : 1 - \frac{1}{v}

= \frac{1}{2} \cdot \int_{4 x^{2} - 4 x + 2}^{2} 1 - \frac{1}{v} d v

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

= \frac{1}{2} (\int_{4 x^{2} - 4 x + 2}^{2} 1 d v - \int_{4 x^{2} - 4 x + 2}^{2} \frac{1}{v} d v)

\int_{4 x^{2} - 4 x + 2}^{2} 1 d v = - 4 x^{2} + 4 x

\int_{4 x^{2} - 4 x + 2}^{2} \frac{1}{v} d v = ln (2) - ln 4 x^{2} - 4 x + 2

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

化简

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

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