展開する ((ln(x^2+y^2)+e^{5x-z})^2)/2

解

展開する

\frac{( ln ( x ^{2} + y ^{2} ) + e ^{5 x - z} ) ^{2}}{2}

\frac{ln ^{2} ( x ^{2} + y ^{2} )}{2} + e^{5 x - z} ln (x^{2} + y^{2}) + \frac{e ^{10 x - 2 z}}{2}

解答ステップ

\frac{( ln ( x ^{2} + y ^{2} ) + e ^{5 x - z} ) ^{2}}{2}

(ln (x^{2} + y^{2}) + e^{5 x - z})^{2} = ln^{2} (x^{2} + y^{2}) + 2 e^{5 x - z} ln (x^{2} + y^{2}) + e^{10 x - 2 z}

= \frac{ln ^{2} ( x ^{2} + y ^{2} ) + 2 e ^{5 x - z} ln ( x ^{2} + y ^{2} ) + e ^{10 x - 2 z}}{2}

分数の規則を適用する:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{ln ^{2} ( x ^{2} + y ^{2} ) + 2 e ^{5 x - z} ln ( x ^{2} + y ^{2} ) + e ^{10 x - 2 z}}{2} = \frac{ln ^{2} ( x ^{2} + y ^{2} )}{2} + \frac{2 e ^{5 x - z} ln ( x ^{2} + y ^{2} )}{2} + \frac{e ^{10 x - 2 z}}{2}

= \frac{ln ^{2} ( x ^{2} + y ^{2} )}{2} + \frac{2 e ^{5 x - z} ln ( x ^{2} + y ^{2} )}{2} + \frac{e ^{10 x - 2 z}}{2}

数を割る:

\frac{2}{2} = 1

= \frac{ln ^{2} ( x ^{2} + y ^{2} )}{2} + e^{5 x - z} ln (x^{2} + y^{2}) + \frac{e ^{10 x - 2 z}}{2}