integral of 3x^3-xsqrt(x^4)+5e^x+3x^{1/3}-x-\sqrt[4]{x}+54

解

\int 3 x^{3} - x x^{4} + 5 e^{x} + 3 x^{\frac{1}{3}} - x - 4 x + 54 d x

\frac{3 x ^{4}}{4} - \frac{x ^{4}}{4} + 5 e^{x} + \frac{9}{4} x^{\frac{4}{3}} - \frac{x ^{2}}{2} - \frac{4}{5} x^{\frac{5}{4}} + 54 x + C

解答ステップ

\int 3 x^{3} - x x^{4} + 5 e^{x} + 3 x^{\frac{1}{3}} - x - 4 x + 54 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 3 x^{3} d x - \int x x^{4} d x + \int 5 e^{x} d x + \int 3 x^{\frac{1}{3}} d x - \int x d x - \int 4 x d x + \int 54 d x

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

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

\int 5 e^{x} d x = 5 e^{x}

\int 3 x^{\frac{1}{3}} d x = \frac{9}{4} x^{\frac{4}{3}}

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

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

\int 54 d x = 54 x

= \frac{3 x ^{4}}{4} - \frac{x ^{4}}{4} + 5 e^{x} + \frac{9}{4} x^{\frac{4}{3}} - \frac{x ^{2}}{2} - \frac{4}{5} x^{\frac{5}{4}} + 54 x

定数を解答に追加する

= \frac{3 x ^{4}}{4} - \frac{x ^{4}}{4} + 5 e^{x} + \frac{9}{4} x^{\frac{4}{3}} - \frac{x ^{2}}{2} - \frac{4}{5} x^{\frac{5}{4}} + 54 x + C