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人気のある微分積分 >

integral of xarctan(asqrt(x))

解

\int x arctan (a x) d x

解

\frac{1}{2} x^{2} arctan (a x) - \frac{1}{6 a ^{4}} (a^{3} x^{\frac{3}{2}} - 3 a x + 3 arctan (a x)) + C

解答ステップ

\int x arctan (a x) d x

u置換積分法を適用する

= \int \frac{2 u ^{3} arctan ( u )}{a ^{4}} d u

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{a ^{4}} \cdot \int u^{3} arctan (u) d u

部分積分を適用する

= \frac{2}{a ^{4}} (\frac{1}{4} u^{4} arctan (u) - \int \frac{u ^{4}}{4 ( u ^{2} + 1 )} d u)

\int \frac{u ^{4}}{4 ( u ^{2} + 1 )} d u = \frac{1}{4} (\frac{u ^{3}}{3} - u + arctan (u))

= \frac{2}{a ^{4}} (\frac{1}{4} u^{4} arctan (u) - \frac{1}{4} (\frac{u ^{3}}{3} - u + arctan (u)))

代用を戻す

u = a x

= \frac{2}{a ^{4}} (\frac{1}{4} (a x)^{4} arctan (a x) - \frac{1}{4} (\frac{( a x ) ^{3}}{3} - a x + arctan (a x)))

簡素化

\frac{2}{a ^{4}} (\frac{1}{4} (a x)^{4} arctan (a x) - \frac{1}{4} (\frac{( a x ) ^{3}}{3} - a x + arctan (a x))) : \frac{1}{2} x^{2} arctan (a x) - \frac{1}{6 a ^{4}} (a^{3} x^{\frac{3}{2}} - 3 a x + 3 arctan (a x))

= \frac{1}{2} x^{2} arctan (a x) - \frac{1}{6 a ^{4}} (a^{3} x^{\frac{3}{2}} - 3 a x + 3 arctan (a x))

定数を解答に追加する

= \frac{1}{2} x^{2} arctan (a x) - \frac{1}{6 a ^{4}} (a^{3} x^{\frac{3}{2}} - 3 a x + 3 arctan (a x)) + C

グラフ

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