integral of (3/(x^{3/4)}-4\sqrt[3]{x}+1)

解

\int (\frac{3}{x ^{\frac{3}{4}}} - 4 3 x + 1) d x

12 4 x - 3 x^{\frac{4}{3}} + x + C

解答ステップ

\int \frac{3}{x ^{\frac{3}{4}}} - 4 3 x + 1 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 \frac{3}{x ^{\frac{3}{4}}} d x - \int 4 3 x d x + \int 1 d x

\int \frac{3}{x ^{\frac{3}{4}}} d x = 12 x^{\frac{1}{4}}

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

\int 1 d x = x

= 12 x^{\frac{1}{4}} - 3 x^{\frac{4}{3}} + x

累乗根の規則を適用する:

a^{\frac{1}{n}} = n a

= 12 4 x - 3 x^{\frac{4}{3}} + x

定数を解答に追加する

= 12 4 x - 3 x^{\frac{4}{3}} + x + C