integral from a to 2a of (ax-(a^3)/x)

解

\int_{a}^{2 a} (a x - \frac{a ^{3}}{x}) d x

a (2 a^{2} - \frac{a ^{2}}{2}) - ln (2) a^{3}

解答ステップ

\int_{a}^{2 a} a x - \frac{a ^{3}}{x} d x

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_{a}^{2 a} a x d x - \int_{a}^{2 a} \frac{a ^{3}}{x} d x

\int_{a}^{2 a} a x d x = a (2 a^{2} - \frac{a ^{2}}{2})

\int_{a}^{2 a} \frac{a ^{3}}{x} d x = ln (2) a^{3}

= a (2 a^{2} - \frac{a ^{2}}{2}) - ln (2) a^{3}