integral of (5x^3+2sqrt(x)+1/(x^3))

解

\int (5 x^{3} + 2 x + \frac{1}{x ^{3}}) d x

\frac{5 x ^{4}}{4} + \frac{4}{3} x^{\frac{3}{2}} - \frac{1}{2 x ^{2}} + C

解答ステップ

\int 5 x^{3} + 2 x + \frac{1}{x ^{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^{3} d x + \int 2 x d x + \int \frac{1}{x ^{3}} d x

\int 5 x^{3} d x = \frac{5 x ^{4}}{4}

\int 2 x d x = \frac{4}{3} x^{\frac{3}{2}}

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

= \frac{5 x ^{4}}{4} + \frac{4}{3} x^{\frac{3}{2}} - \frac{1}{2 x ^{2}}

定数を解答に追加する

= \frac{5 x ^{4}}{4} + \frac{4}{3} x^{\frac{3}{2}} - \frac{1}{2 x ^{2}} + C