+101 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.
+37157 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
+37157 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
+9674 \(\quad\displaystyle \int \sqrt x \left(1-x^2\right)\;\mathbb{d}x\\ \text{Substitute }u^2 = x\text{.}\\ =\displaystyle \int u \left(1-u^4\right) (2u) \;\mathbb{d}u\\ =2 \displaystyle \int \left(u^2 - u^6\right) \;\mathbb{d}u\\ = \dfrac{2u^3}{3} - \dfrac{2u^7}{7} + \mathbf{C}\\ = \dfrac{2x\sqrt x}{3}-\dfrac{2x^3\sqrt x}{7} + \mathbf C\)
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