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How do you evaluate the integral #int dx/(2x-7)#?

Answer 1

Scarlett Allison

#int dx/(2x-7) = 1/2 int (d(2x-7))/(2x-7) = 1/2ln abs (2x-7) +C#

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Answer 2

Aurora Alvarado

To evaluate the integral ( \int \frac{dx}{2x - 7} ), we can use the technique of u-substitution. Let ( u = 2x - 7 ), then ( du = 2dx ). Solving for ( dx ), we get ( dx = \frac{du}{2} ). Substituting these into the integral, we have:

[ \int \frac{dx}{2x - 7} = \int \frac{du}{2 \cdot u} ]

This simplifies to:

[ \frac{1}{2} \int \frac{du}{u} ]

Now, integrating ( \frac{1}{u} ) with respect to ( u ), we get:

[ \frac{1}{2} \ln|u| + C ]

Substituting back ( u = 2x - 7 ), we have:

[ \frac{1}{2} \ln|2x - 7| + C ]

Therefore, the integral of ( \frac{dx}{2x - 7} ) is ( \frac{1}{2} \ln|2x - 7| + 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.