How do you evaluate the integral #int tan theta d(theta)# from 0 to #pi/2#?

Answer 1

integral non convergent :-(

#int_0^(pi/2) d theta tan theta#
#= int_0^(pi/2) d theta sin theta / cos theta#
and because #d/dx ln f(x) = 1/f(x) f'(x)#
#= - [ln cos theta ]_0^(pi/2)#
#= [ln cos theta ]_(pi/2)^0#
#= [ln cos 0 ] - [ln cos( pi/2) ]#
#= [ln 1 ] - [color{red}{ln 0} ]#
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Answer 2

To evaluate the integral (\int \tan(\theta) , d\theta) from (0) to (\frac{\pi}{2}), we use the following steps:

  1. Recognize that (\int \tan(\theta) , d\theta) is an improper integral due to the singularity of (\tan(\theta)) at (\frac{\pi}{2}).
  2. Apply the substitution method: Let (u = \tan(\theta)), then (du = \sec^2(\theta) , d\theta).
  3. Rewrite the integral in terms of (u): (\int u , du).
  4. Integrate (u) with respect to (u).
  5. Reverse the substitution to find the final result.

Following these steps:

  1. Substitute (u = \tan(\theta)), then (du = \sec^2(\theta) , d\theta).
  2. The limits of integration change accordingly: when (\theta = 0), (u = \tan(0) = 0), and when (\theta = \frac{\pi}{2}), (u = \tan(\frac{\pi}{2})) approaches infinity.
  3. Rewrite the integral in terms of (u): (\int u , du).
  4. Integrate (u) with respect to (u): (\frac{1}{2}u^2).
  5. Apply the limits of integration: (\frac{1}{2}(\tan(\frac{\pi}{2}))^2 - \frac{1}{2}(0)^2).
  6. Since (\tan(\frac{\pi}{2})) approaches infinity, the integral is divergent.

Therefore, the integral (\int_0^{\frac{\pi}{2}} \tan(\theta) , d\theta) diverges.

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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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