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

integral non convergent :-(

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

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

Following these steps:

- Substitute (u = \tan(\theta)), then (du = \sec^2(\theta) , d\theta).
- 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.
- Rewrite the integral in terms of (u): (\int u , du).
- Integrate (u) with respect to (u): (\frac{1}{2}u^2).
- Apply the limits of integration: (\frac{1}{2}(\tan(\frac{\pi}{2}))^2 - \frac{1}{2}(0)^2).
- 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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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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