integral of (x^3)/(sqrt(36x^2+1296))

解

\int \frac{x ^{3}}{36 x ^{2} + 1296} d x

\frac{x ^{4}}{6 x ^{2} + 36} - \frac{1}{3} (\frac{1}{3} (x^{2} + 36)^{\frac{3}{2}} - 18 x^{2} + 36 + \frac{648}{x ^{2} + 36}) + C

解答ステップ

\int \frac{x ^{3}}{36 x ^{2} + 1296} d x

部分積分を適用する

= \frac{x ^{4}}{6 x ^{2} + 36} - \int \frac{x ^{3} ( x ^{2} + 54 )}{3 ( x ^{2} + 36 ) x ^{2} + 36} d x

\int \frac{x ^{3} ( x ^{2} + 54 )}{3 ( x ^{2} + 36 ) x ^{2} + 36} d x = \frac{1}{3} (\frac{1}{3} (x^{2} + 36)^{\frac{3}{2}} - 18 x^{2} + 36 + \frac{648}{x ^{2} + 36})

= \frac{x ^{4}}{6 x ^{2} + 36} - \frac{1}{3} (\frac{1}{3} (x^{2} + 36)^{\frac{3}{2}} - 18 x^{2} + 36 + \frac{648}{x ^{2} + 36})

定数を解答に追加する

= \frac{x ^{4}}{6 x ^{2} + 36} - \frac{1}{3} (\frac{1}{3} (x^{2} + 36)^{\frac{3}{2}} - 18 x^{2} + 36 + \frac{648}{x ^{2} + 36}) + C