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How do you find the antiderivative of #2x^(-2/3) dx#?

Answer 1

#6x^(1/3)+C#

We know that, #color(red)(intx^ndx=x^(n+1)/(n+1)+C)# #rArrint2x^-(2/3)dx=2(x^(-2/3+1)/(-2/3+1))+C=2(x^((-2+3)/3)/((-2+3)/3))+C=2(x^(1/3)/(1/3))+C=6x^(1/3)+C#

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Answer 2

Axel Alvarez

To find the antiderivative of (2x^{-\frac{2}{3}}) dx, you can use the power rule for antiderivatives. The power rule states that if (f(x) = x^n), then the antiderivative of (f(x)) with respect to (x) is (\frac{x^{n+1}}{n+1} + C), where (C) is the constant of integration. Applying this rule to (2x^{-\frac{2}{3}}), we get:

[\int 2x^{-\frac{2}{3}} , dx = \frac{2x^{-\frac{2}{3} + 1}}{-\frac{2}{3} + 1} + C]

Simplify the exponent and denominator:

[\int 2x^{-\frac{2}{3}} , dx = \frac{2x^{\frac{1}{3}}}{\frac{1}{3}} + C]

Now, simplify further:

[\int 2x^{-\frac{2}{3}} , dx = 6x^{\frac{1}{3}} + C]

So, the antiderivative of (2x^{-\frac{2}{3}}) dx is (6x^{\frac{1}{3}} + C), where (C) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.