integral of (5xsqrt(x)-1/(x^3)+3)

解

\int (5 x x - \frac{1}{x ^{3}} + 3) d x

2 x^{\frac{5}{2}} + \frac{1}{2 x ^{2}} + 3 x + C

解答ステップ

\int 5 x x - \frac{1}{x ^{3}} + 3 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 5 x x d x - \int \frac{1}{x ^{3}} d x + \int 3 d x

\int 5 x x d x = 2 x^{\frac{5}{2}}

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

\int 3 d x = 3 x

= 2 x^{\frac{5}{2}} - (- \frac{1}{2 x ^{2}}) + 3 x

簡素化

= 2 x^{\frac{5}{2}} + \frac{1}{2 x ^{2}} + 3 x

定数を解答に追加する

= 2 x^{\frac{5}{2}} + \frac{1}{2 x ^{2}} + 3 x + C