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# When you want to do antiderivatives but your brain goes antiwork

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Ok so I have this antiderivative problem I am confused about.

So it is to get the antiderivative of $$\int \sqrt{x}(1-{x}^{2})dx$$

Please give me the steps and explanation of how you got to the answer.

Jan 12, 2019

#1
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Maybe something like this?

x^1/2 ( 1-x^2) =   x^1/2 - x^5/2

Integrate       2/3 x^3/2 - 2/7 x^7/2 + C       where C = a constant

Jan 12, 2019

#1
+19709
0

Maybe something like this?

x^1/2 ( 1-x^2) =   x^1/2 - x^5/2

Integrate       2/3 x^3/2 - 2/7 x^7/2 + C       where C = a constant

ElectricPavlov Jan 12, 2019
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$$\quad\displaystyle \int \sqrt x \left(1-x^2\right)\;\mathbb{d}x\\ \text{Substitute }u^2 = x\text{.}\\ =\displaystyle \int u \left(1-u^4\right) (2u) \;\mathbb{d}u\\ =2 \displaystyle \int \left(u^2 - u^6\right) \;\mathbb{d}u\\ = \dfrac{2u^3}{3} - \dfrac{2u^7}{7} + \mathbf{C}\\ = \dfrac{2x\sqrt x}{3}-\dfrac{2x^3\sqrt x}{7} + \mathbf C$$

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Jan 14, 2019