For what values of a is each integral improper? \int_{a}^{4}\frac{x}{3x-1}dx

ajedrezlaproa6j

ajedrezlaproa6j

Answered question

2021-12-31

For what values of a is each integral improper?
a4x3x1dx

Answer & Explanation

Jonathan Burroughs

Jonathan Burroughs

Beginner2022-01-01Added 37 answers

Consider the integral a4x3x1dx.
Check whether the provided function is improper or not, find the point of infinite discontinuity.
Equate the denominator to zero.
3x-1=0
3x=1
x=13
The function has a discontinuity at x=13, so the integral is improper on the interval [a,4] for a13.
And the integral is improper when a=
Thus, the given integral is improper for a13 or a=.
Matthew Rodriguez

Matthew Rodriguez

Beginner2022-01-02Added 32 answers

a4x3x1dx=a413(u+1)3udu
=133a4u+1udu
=19(a41udu+a41du)...(4)
=19[[ln|u|]a4+[u]a4]
=19{ln11ln(3a1)+[123a]}
=19[2.398ln3a+123a]
=19[14.398ln3a3a]
Now is
1.5998=19[ln3a+3a]
14.398=ln3a+3a
a=4
Vasquez

Vasquez

Expert2022-01-08Added 669 answers

Recall that an integral is improper if one of the limits is ± or it has infinite discontinuities in the interval of integration. In this example, value a is a value at which graph of y=x3x1 is discontinuous. We can easily determine that graphs of rational functions have a discontinuity when the denominator is equal to 0.
3x1=0x=13
We determined that the integral is improper for a=13

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