integral from 3 to 6 of (6x+9)ln(x+3)

解答

\int_{3}^{6} (6 x + 9) ln (x + 3) d x

162 ln (3) + 9 ln (\frac{531441}{64}) - \frac{81}{2}

十进制

218.69537 \dots

求解步骤

\int_{3}^{6} (6 x + 9) ln (x + 3) d x

乘开

(6 x + 9) ln (x + 3) : 6 x ln (x + 3) + 9 ln (x + 3)

= \int_{3}^{6} 6 x ln (x + 3) + 9 ln (x + 3) d x

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_{3}^{6} 6 x ln (x + 3) d x + \int_{3}^{6} 9 ln (x + 3) d x

\int_{3}^{6} 6 x ln (x + 3) d x = 6 (27 ln (3) - \frac{9}{4})

\int_{3}^{6} 9 ln (x + 3) d x = 9 (ln (\frac{531441}{64}) - 3)

= 6 (27 ln (3) - \frac{9}{4}) + 9 (ln (\frac{531441}{64}) - 3)

化简

6 (27 ln (3) - \frac{9}{4}) + 9 (ln (\frac{531441}{64}) - 3) : 162 ln (3) + 9 ln (\frac{531441}{64}) - \frac{81}{2}

= 162 ln (3) + 9 ln (\frac{531441}{64}) - \frac{81}{2}

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