integral from 3 to 6 of (2ax+3)3x

解

\int_{3}^{6} (2 a x + 3) 3 x d x

3 (126 a + \frac{81}{2})

解答ステップ

\int_{3}^{6} (2 a x + 3) \cdot 3 x d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int_{3}^{6} (2 a x + 3) x d x

拡張

(2 a x + 3) x : 2 a x^{2} + 3 x

= 3 \cdot \int_{3}^{6} 2 a x^{2} + 3 x 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

= 3 (\int_{3}^{6} 2 a x^{2} d x + \int_{3}^{6} 3 x d x)

\int_{3}^{6} 2 a x^{2} d x = 126 a

\int_{3}^{6} 3 x d x = \frac{81}{2}

= 3 (126 a + \frac{81}{2})