integral from 1 to 2 of (v^3+3v^6)/(v^4)

解

\int_{1}^{2} \frac{v ^{3} + 3 v ^{6}}{v ^{4}} d v

ln (2) + 7

十進法表記

7.69314 \dots

解答ステップ

\int_{1}^{2} \frac{v ^{3} + 3 v ^{6}}{v ^{4}} d v

拡張

\frac{v ^{3} + 3 v ^{6}}{v ^{4}} : \frac{1}{v} + 3 v^{2}

= \int_{1}^{2} \frac{1}{v} + 3 v^{2} d v

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{2} \frac{1}{v} d v + \int_{1}^{2} 3 v^{2} d v

\int_{1}^{2} \frac{1}{v} d v = ln (2)

\int_{1}^{2} 3 v^{2} d v = 7

= ln (2) + 7