integral from 2y to 3-y of (x+y)

解答

\int_{2 y}^{3 - y} (x + y) d x

- 5 y^{2} + 3 y + \frac{( - y + 3 ) ^{2}}{2}

求解步骤

\int_{2 y}^{3 - y} x + y d x

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_{2 y}^{3 - y} y d x + \int_{2 y}^{3 - y} x d x

\int_{2 y}^{3 - y} y d x = 3 y - 3 y^{2}

\int_{2 y}^{3 - y} x d x = \frac{( 3 - y ) ^{2}}{2} - 2 y^{2}

= 3 y - 3 y^{2} + \frac{( 3 - y ) ^{2}}{2} - 2 y^{2}

化简

= - 5 y^{2} + 3 y + \frac{( - y + 3 ) ^{2}}{2}

\int_{1}^{5} (3 + \frac{2}{x}) d x

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