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How do you integrate #f(x)=(sqrtx-1)(sqrtx+1)# using the product rule?

Answer 1

Evelyn Agard

#1/2x^2-x+C#

Use identity #(a+b)(a-b)=a^2-b^2#

In our case #(sqrtx-1)(sqrtx+1)=x-1#

#int(x-1)dx=intxdx-intdx=1/2x^2-x+C#

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Answer 2

Scarlett Allison

To integrate ( f(x) = (\sqrt{x} - 1)(\sqrt{x} + 1) ) using the product rule, we first expand the expression: [ f(x) = \sqrt{x} \cdot \sqrt{x} + \sqrt{x} \cdot 1 - 1 \cdot \sqrt{x} - 1 \cdot 1 ]

[ = x + \sqrt{x} - \sqrt{x} - 1 ]

[ = x - 1 ]

Now, we can integrate ( f(x) ): [ \int f(x) , dx = \int (x - 1) , dx ]

[ = \frac{x^2}{2} - x + 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.