integral from 2x^2 to x^4 of 2t+t^2

解

\int_{2 x^{2}}^{x^{4}} 2 t + t^{2} d t

x^{8} - 4 x^{4} + \frac{x ^{12} - 8 x ^{6}}{3}

解答ステップ

\int_{2 x^{2}}^{x^{4}} 2 t + t^{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_{2 x^{2}}^{x^{4}} 2 t d t + \int_{2 x^{2}}^{x^{4}} t^{2} d t

\int_{2 x^{2}}^{x^{4}} 2 t d t = x^{8} - 4 x^{4}

\int_{2 x^{2}}^{x^{4}} t^{2} d t = \frac{x ^{12} - 8 x ^{6}}{3}

= x^{8} - 4 x^{4} + \frac{x ^{12} - 8 x ^{6}}{3}