How do you evaluate the integral #int 1/sqrt(1-x)dx# from 0 to 1?

Answer 1

Use integration by substitution (u-substitution).

We can evaluate this integral using integration by substitution, or u-substitution. We pick some part of the integrand to set equal to some variable (such as #u#, but any variable is an option). Good places to look at first include under a radical or in the denominator. This is not always the case, but it is in this one.
We can set #u=1-x#

Therefore,

#du=-1dx# #-du=dx#

We can substitute these values into our integral. We get:

#-int1/sqrtudu#

Which we can rewrite as:

#-intu^(-1/2)du#

Integrating, we get:

#-2u^(1/2)#
From here you have two options on evaluating for the given limits of integration. You can either choose now to substitute #1-x# back in for #u# and evaluate from 0 to 1, or you can change the limits of integration and evaluate with u. I will demonstrate both options.
#-2(1-x)^(1/2)# #-2[(1-1)^(1/2)-(1-0)^(1/2)]# #-2(-1)#

Final answer: 2

#u=1-x#
#u=1-(1) # #u=0# (new upper limit)
#u=1-0# #u=1# (new lower limit)

Evaluating, we have

#-2[(0)^(1/2)-(1)^(1/2)]# #-2(-1)#

Final answer: 2

Hope this helps!

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Answer 2

To evaluate ( \int_{0}^{1} \frac{1}{\sqrt{1-x}} , dx ), we can make the substitution ( u = 1 - x ). This changes the limits of integration to ( u(0) = 1 ) and ( u(1) = 0 ), and the integral becomes:

[ \int_{1}^{0} \frac{1}{\sqrt{u}} , du ]

which equals ( 2 ).

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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.

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.

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.

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.

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