Evaluate the integral or show that it is divergent?

Answer 1

The improper integral converges and equals #32/3#.

First note that

#int dx/(x+2)^(1/4)=int (x+2)^(-1/4)dx=4/3 (x+2)^(3/4)+C#.
Next, note that the "impropriety" in the definite integral occurs at #x=-2# because the function has a vertical asymptote there. Hence,
#int_{-2}^{14}dx/(x+2)^(1/4)=lim_{a->-2+}int_{a}^{14}dx/(x+2)^(1/4)#
assuming this limit exists, where the positive sign to the right of #-2# in the limit indicates that #a# is approaching #-2# from the right.

By the Fundamental Theorem of Calculus,

#int_{a}^{14}dx/(x+2)^(1/4)=4/3 (x+2)^(3/4)|_{a}^{14}#
#=4/3 * 16^(3/4) - 4/3(a+2)^(3/4)=32/3 - 4/3(a+2)^(3/4)#.
Since #lim_{a->-2+}(a+2)^(3/4)=0^(3/4)=0#, it follows that the improper integral converges to #32/3#.

We therefore write

#int_{-2}^{14}dx/(x+2)^(1/4)=32/3#.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7