integral from 1 to e^2 of ((in(x))^2)/x

解

\int_{1}^{e^{2}} \frac{( in ( x ) ) ^{2}}{x} d x

\frac{e ^{4} i ^{2} n ^{2} - i ^{2} n ^{2}}{2}

解答ステップ

\int_{1}^{e^{2}} \frac{( in x ) ^{2}}{x} d x

部分積分を適用する

= [i^{2} n^{2} x^{2} - \int i^{2} n^{2} x d x]_{1}^{e^{2}}

\int i^{2} n^{2} x d x = \frac{i ^{2} n ^{2} x ^{2}}{2}

= [i^{2} n^{2} x^{2} - \frac{i ^{2} n ^{2} x ^{2}}{2}]_{1}^{e^{2}}

結合

i^{2} n^{2} x^{2} - \frac{i ^{2} n ^{2} x ^{2}}{2} : \frac{i ^{2} n ^{2} x ^{2}}{2}

= [\frac{i ^{2} n ^{2} x ^{2}}{2}]_{1}^{e^{2}}

限度を計算する:

\frac{e ^{4} i ^{2} n ^{2}}{2} - \frac{i ^{2} n ^{2}}{2}

= \frac{e ^{4} i ^{2} n ^{2}}{2} - \frac{i ^{2} n ^{2}}{2}

簡素化

= \frac{e ^{4} i ^{2} n ^{2} - i ^{2} n ^{2}}{2}