integral of ((x^2-4))/(x(x^2+4))

解

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

ln x^{2} + 4 - ln ∣ x ∣ + C

解答ステップ

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

拡張

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

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

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

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

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

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

簡素化

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

= ln x^{2} + 4 - ln ∣ x ∣

定数を解答に追加する

= ln x^{2} + 4 - ln ∣ x ∣ + C