integral from 0 to 6-2x of (6-2x-y)

解答

\int_{0}^{6 - 2 x} (6 - 2 x - y) d y

4 x^{2} - 24 x - \frac{( - 2 x + 6 ) ^{2}}{2} + 36

求解步骤

\int_{0}^{6 - 2 x} 6 - 2 x - y d y

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{6 - 2 x} 6 d y - \int_{0}^{6 - 2 x} y d y - \int_{0}^{6 - 2 x} 2 x d y

\int_{0}^{6 - 2 x} 6 d y = 6 (- 2 x + 6)

\int_{0}^{6 - 2 x} y d y = \frac{( - 2 x + 6 ) ^{2}}{2}

\int_{0}^{6 - 2 x} 2 x d y = 2 x (- 2 x + 6)

= 6 (- 2 x + 6) - \frac{( - 2 x + 6 ) ^{2}}{2} - 2 x (- 2 x + 6)

化简

6 (- 2 x + 6) - \frac{( - 2 x + 6 ) ^{2}}{2} - 2 x (- 2 x + 6) : 4 x^{2} - 24 x - \frac{( - 2 x + 6 ) ^{2}}{2} + 36

= 4 x^{2} - 24 x - \frac{( - 2 x + 6 ) ^{2}}{2} + 36

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