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受欢迎的微积分 >

integral of sqrt(x^4-2x^3+x^2)

解答

\int x^{4} - 2 x^{3} + x^{2} d x

解答

\frac{x ^{3}}{3} - \frac{x ^{2}}{2} + C

求解步骤

\int x^{4} - 2 x^{3} + x^{2} d x

分解

x^{4} - 2 x^{3} + x^{2} : x^{2} (x - 1)^{2}

= \int x^{2} (x - 1)^{2} d x

使用指数法则:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x^{2} (x - 1)^{2} = x^{2} (x - 1)^{2}

= \int x^{2} (x - 1)^{2} d x

使用指数法则:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x^{2} (x - 1)^{2} = (x (x - 1))^{2}

= \int (x (x - 1))^{2} d x

(x (x - 1))^{2} = (x (x - 1)),

假定

x (x - 1) \geq 0

= \int x (x - 1) d x

乘开

x (x - 1) : x^{2} - x

= \int x^{2} - x 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 x^{2} d x - \int x d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int x d x = \frac{x ^{2}}{2}

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2}

解答补常数

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + C

作图

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d er i v a t i v e F (x) = (3 x^{6} + 4 x^{3})^{4}

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\frac{d y}{d t} = - 4 y + 9 e^{- t}

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