integral of ((7x-4)(3x+5)^2)/(x^2)

解

\int \frac{( 7 x - 4 ) ( 3 x + 5 ) ^{2}}{x ^{2}} d x

\frac{63 x ^{2}}{2} + 174 x + 55 ln ∣ x ∣ + \frac{100}{x} + C

解答ステップ

\int \frac{( 7 x - 4 ) ( 3 x + 5 ) ^{2}}{x ^{2}} d x

拡張

\frac{( 7 x - 4 ) ( 3 x + 5 ) ^{2}}{x ^{2}} : 63 x + 174 + \frac{55}{x} - \frac{100}{x ^{2}}

= \int 63 x + 174 + \frac{55}{x} - \frac{100}{x ^{2}} d x

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 63 x d x + \int 174 d x + \int \frac{55}{x} d x - \int \frac{100}{x ^{2}} d x

\int 63 x d x = \frac{63 x ^{2}}{2}

\int 174 d x = 174 x

\int \frac{55}{x} d x = 55 ln ∣ x ∣

\int \frac{100}{x ^{2}} d x = - \frac{100}{x}

= \frac{63 x ^{2}}{2} + 174 x + 55 ln ∣ x ∣ - (- \frac{100}{x})

簡素化

= \frac{63 x ^{2}}{2} + 174 x + 55 ln ∣ x ∣ + \frac{100}{x}

定数を解答に追加する

= \frac{63 x ^{2}}{2} + 174 x + 55 ln ∣ x ∣ + \frac{100}{x} + C