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What is the antiderivative of #xsqrtx#?

Answer 1

# 2/5 x^(5/2) #

Note : # x sqrt(x) = x x^(1/2) = x^(1 + 1/2) = x^(3/2) # Therefore, # int x sqrt(x) dx = int x^(3/2) dx = (x^(3/2 + 1)) / (3/2 + 1) = 2/5 x^(5/2) #

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Answer 2

Emma Aldridge

You can simply multiply them together (more explicitly).

#xsqrtx = x^("3/2")#

And then just use the reverse Power Rule.

#d/(dx)[x^("3/2")] = 2/5x^("5/2")#

Then, since an antiderivative is a generalization of what an integral does, they are almost the same thing. Therefore, we add a constant to imply that you get every single function that is within the antiderivative's slope field.

(notice the various vertical-shift variations of a single function forms the slope field)

#-> color(blue)(2/5x^("5/2") + C)#

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Answer 3

Charles Arocha

The antiderivative of ( x\sqrt{x} ) is ( \frac{2}{5}x^\frac{5}{2} + 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.