How do you test the improper integral $\int \frac{sin θ}{{cos}^{2} θ}$ $int sintheta/cos^2theta$ from $[0, \frac{π}{2}]$ $[0,pi/2]$ and evaluate if possible?

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How do you test the improper integral $\int \frac{sin θ}{{cos}^{2} θ}$ $int sintheta/cos^2theta$ from $[0, \frac{π}{2}]$ $[0,pi/2]$ and evaluate if possible?

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Answer 1 · 2017-08-14T17:15:01

Use limits.

Explanation:

The integral is improper where $cos θ = 0$ $costheta = 0$ , which occurs at $θ = \frac{π}{2}$ $theta = pi/2$ .
To evaluate,
$\int_{0}^{\frac{π}{2}} \frac{sin θ}{{cos}^{2} θ} d (θ)$ $int_0^(pi/2) sintheta/cos^2theta d(theta)$
first observe that $\frac{sin θ}{{cos}^{2} θ} = sec (θ) tan (θ)$ $sintheta/cos^2theta = sec(theta)tan(theta)$
Use a limit on the improper end:
$int_0^(pi/2) sec(theta)tan(theta) d(theta) = lim(int_0^bsec(theta)tan(theta) d(theta) )$ , where the limit is as $b -> (pi/2)^-$
Evaluate:
$lim(int_0^b sec(theta)tan(theta) d(theta) ) =$
$= lim(secb - sec0)$
$= lim(secb - 1)$
This limit diverges as $b -> pi/2$ .

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How do you test the improper integral $\int \frac{sin θ}{{cos}^{2} θ}$ $int sintheta/cos^2theta$ from $[0, \frac{π}{2}]$ $[0,pi/2]$ and evaluate if possible?

1 Answer

Explanation:

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How do you test the improper integral int sintheta/cos^2theta∫sinθcos2θint sintheta/cos^2theta from [0,pi/2][0,π2][0,pi/2] and evaluate if possible?

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Explanation:

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How do you test the improper integral $\int \frac{sin θ}{{cos}^{2} θ}$ $int sintheta/cos^2theta$ from $[0, \frac{π}{2}]$ $[0,pi/2]$ and evaluate if possible?