What is the integral of #int sec(x)# from 0 to 2?

Answer 1

That improper integral diverges (does not exist).

Because #secx# is not defined at #pi/2# which is less than #2#, the integral is improper. Improper integral are probably not suitable for an "Introduction to Integration". In the early portion of a course on integration, the definite integral is often defined for function defined on some closed interval #[a,b]#. If this is the only definition you have so far, then the appropriate answer is, "the integral is not defined". (Meaning, "is not defined, yet".)

After getting a definition of improper integral, we would try to find this integral by evaluating both

#int_0^(pi/2) secx dx# and #int_(pi/2)^2 secx dx#.
#int_0^(pi/2) secx dx = lim_(brarr(pi/2) ^-) int_0^b secx dx#
# = lim_(brarr(pi/2)^-) ln abs(tanx+secx)]_0^b#
# = lim_(brarr(pi/2)^-) (ln abs(tanb+secb)-ln abs(tan0 + sec0))#
# = lim_(brarr(pi/2)^-) ln abs(tanb+secb)#
As #brarr(pi/2)^-#, both #tanb# and #secb# increase without bound, so their sum increases without bound. Therefore,
# lim_(brarr(pi/2)^-) ln abs(tanb+secb)= oo#.

That is, the integral diverges.

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Answer 2

To find the integral of sec(x) from 0 to 2, you can use the integral formula:

∫ sec(x) dx = ln|sec(x) + tan(x)| + C

Integrating sec(x) from 0 to 2:

∫ sec(x) dx = ln|sec(2) + tan(2)| - ln|sec(0) + tan(0)|

sec(0) = 1, tan(0) = 0, sec(2) and tan(2) are calculated using trigonometric functions.

Then substitute the values into the equation and calculate the result.

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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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