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人気のある微分積分 >

integral of (2(x^2+1))/(x(x^2+2))

解

\int \frac{2 ( x ^{2} + 1 )}{x ( x ^{2} + 2 )} d x

解

\frac{1}{2} ln x^{2} + 2 + ln (x) + C

解答ステップ

\int \frac{2 ( x ^{2} + 1 )}{x ( x ^{2} + 2 )} d x

定数を除く:

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

= 2 \cdot \int \frac{x ^{2} + 1}{x ( x ^{2} + 2 )} d x

拡張

\frac{x ^{2} + 1}{x ( x ^{2} + 2 )} : \frac{x ^{2}}{x ( x ^{2} + 2 )} + \frac{1}{x ( x ^{2} + 2 )}

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{x ^{2}}{x ( x ^{2} + 2 )} d x + \int \frac{1}{x ( x ^{2} + 2 )} d x)

\int \frac{x ^{2}}{x ( x ^{2} + 2 )} d x = \frac{1}{2} ln x^{2} + 2

\int \frac{1}{x ( x ^{2} + 2 )} d x = - \frac{1}{4} (ln x^{2} + 2 - 2 ln (x))

= 2 (\frac{1}{2} ln x^{2} + 2 - \frac{1}{4} (ln x^{2} + 2 - 2 ln (x)))

簡素化

2 (\frac{1}{2} ln x^{2} + 2 - \frac{1}{4} (ln x^{2} + 2 - 2 ln (x))) : \frac{1}{2} ln x^{2} + 2 + ln (x)

= \frac{1}{2} ln x^{2} + 2 + ln (x)

定数を解答に追加する

= \frac{1}{2} ln x^{2} + 2 + ln (x) + C

グラフ

人気の例

limit as x approaches 4-of 1/(x^2-16)

x \to 4 - lim (\frac{1}{x ^{2} - 16})

integral of x/(sqrt((1-x^2)))

\int \frac{x}{( 1 - x ^{2} )} d x

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\frac{d}{d x} (x^{5 c o s (x)})