integral of ra^r

解

\int r a^{r} d r

\frac{a ^{r} r}{ln ( a )} - \frac{a ^{r}}{ln ^{2} ( a )} + C

解答ステップ

\int r a^{r} d r

部分積分を適用する

: \frac{a ^{r} r}{ln ( a )} - \int \frac{a ^{r}}{ln ( a )} d r

= \frac{a ^{r} r}{ln ( a )} - \int \frac{a ^{r}}{ln ( a )} d r

\int \frac{a ^{r}}{ln ( a )} d r = \frac{a ^{r}}{ln ^{2} ( a )}

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

定数を解答に追加する

= \frac{a ^{r} r}{ln ( a )} - \frac{a ^{r}}{ln ^{2} ( a )} + C