integral from 0 to x of (t^3+3t^2+2)

解

\int_{0}^{x} (t^{3} + 3 t^{2} + 2) d t

\frac{x ^{4}}{4} + x^{3} + 2 x

解答ステップ

\int_{0}^{x} t^{3} + 3 t^{2} + 2 d t

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_{0}^{x} t^{3} d t + \int_{0}^{x} 3 t^{2} d t + \int_{0}^{x} 2 d t

\int_{0}^{x} t^{3} d t = \frac{x ^{4}}{4}

\int_{0}^{x} 3 t^{2} d t = x^{3}

\int_{0}^{x} 2 d t = 2 x

= \frac{x ^{4}}{4} + x^{3} + 2 x