Integration

This rule of integration $\displaystyle\int \limits^{a}_{0}f\left( x\right) dx =\displaystyle\int \limits^{a}_{0}f\left( a - x\right) dx$ I tried by using this rule to solve this integration but it wasn't solve.. the integration is $\displaystyle\int \limits^{\pi }_{0}\frac{x}{2 - \tan \nolimits^{2}\left( x\right) } dx$ ... I tried wolfram alpha also but it was useless ..... How we can solve this integration by using this rule? . Thanks

#Calculus #Integration #IntegrationTechniques

Easy Math Editor

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Comments

you can apply the rule then add the new integral to the original one, and divide by 2. You will be left with the integral: $\displaystyle \frac{\pi}{2} \int_0^{\pi} \frac{1}{2-\tan^2x} dx$ .

Did you try that??

Hasan Kassim - 5 years, 10 months ago

What will happen if we solve $2 - tan^{2} x = 0$ $x \in (0, \pi)$ ?.....

David Danial - 5 years, 10 months ago

yes there is discontinueties in the integral at :

$\arctan (\sqrt{2} )$ which is between $\frac{\pi}{4}$ and $\frac{\pi}{2}$ .

And $\pi - \arctan (\sqrt{2} )$ .

Hasan Kassim - 5 years, 10 months ago

@Hasan Kassim – So this integration is divergent.... could you please explain more this part of discontinuous.?

David Danial - 5 years, 10 months ago

@David Danial – Yes the integral diverges: lets take take a part of it :

$\displaystyle \int_0^{\arctan (\sqrt{2})} \frac{dx}{2-\tan^2x}$

$\displaystyle = \lim_{a \to \arctan (\sqrt{2})} \int_0^{a} \frac{dx}{2-\tan^2x}$

Use the substitution $u=\tan x$ :

$\displaystyle = \lim_{a \to \arctan (\sqrt{2})} \int_0^{\tan a } \frac{du}{(2-u^2)(1+u^2)}$

Easy by applying partial fractions then integrating, then apply the limit and it will yield a value of infinty. So the whole Integral diverges.

Hasan Kassim - 5 years, 10 months ago

@Trevor B. You might like this!

Calvin Lin Staff - 5 years, 10 months ago

Yes, the Bounds Trick works nicely here. I know how to get to the indefinite, I just need to make the time to write it down and check the bounds. Partial fracs with what Hasan said originally should make it work.

Trevor B. - 5 years, 10 months ago

you could try with partial fractions too

Hjalmar Orellana Soto - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$