integral of 21e^{42x}arctan(e^{21x})

解

\int 21 e^{42 x} arctan (e^{21 x}) d x

21 (\frac{1}{42} e^{42 x} arctan (e^{21 x}) - \frac{1}{42} (- arctan (e^{21 x}) + e^{21 x})) + C

解答ステップ

\int 21 e^{42 x} arctan (e^{21 x}) d x

定数を除く:

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

= 21 \cdot \int e^{42 x} arctan (e^{21 x}) d x

部分積分を適用する

= 21 (\frac{1}{42} e^{42 x} arctan (e^{21 x}) - \int \frac{e ^{63 x}}{2 ( e ^{42 x} + 1 )} d x)

\int \frac{e ^{63 x}}{2 ( e ^{42 x} + 1 )} d x = \frac{1}{42} (- arctan (e^{21 x}) + e^{21 x})

= 21 (\frac{1}{42} e^{42 x} arctan (e^{21 x}) - \frac{1}{42} (- arctan (e^{21 x}) + e^{21 x}))

定数を解答に追加する

= 21 (\frac{1}{42} e^{42 x} arctan (e^{21 x}) - \frac{1}{42} (- arctan (e^{21 x}) + e^{21 x})) + C