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The Proof is Trivial!

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Reply 2520
Problem 413*

Find

0π211+tanα(x)dx.\displaystyle \int_0^\frac{\pi}{2} \frac{1}{1+\tan^\alpha(x)}dx.
Original post by henpen
Problem 413*

Find

0π211+tanα(x)dx.\displaystyle \int_0^\frac{\pi}{2} \frac{1}{1+\tan^\alpha(x)}dx.


pi/4
Original post by henpen
Problem 413*

Find

0π211+tanα(x)dx.\displaystyle \int_0^\frac{\pi}{2} \frac{1}{1+\tan^\alpha(x)}dx.


Solution 413*

Spoiler

(edited 10 years ago)
Reply 2523
Original post by henpen
Problem 413*

Find

0π211+tanα(x)dx.\displaystyle \int_0^\frac{\pi}{2} \frac{1}{1+\tan^\alpha(x)}dx.

​Solution 413
I=0π211+tanα(x)dxI=0π2cosα(x)cosα(x)+sinα(x)dx=0π2sinα(x)cosα(x)+sinα(x)dx2I=0π2dx=π2I=π4I=\displaystyle \int_0^\frac{\pi}{2} \frac{1}{1+\tan^\alpha(x)}dx \\ \\ I=\displaystyle \int_0^\frac{\pi}{2} \frac{\cos^\alpha(x)}{ \cos^\alpha(x)+\sin^\alpha(x)}dx= \displaystyle \int_0^\frac{\pi}{2} \frac{\sin^\alpha(x)}{ \cos^\alpha(x)+\sin^\alpha(x)}dx \\ \\ 2I=\displaystyle \int_0^\frac{\pi}{2}dx= \dfrac{\pi}{2} \\ \\ I=\dfrac{\pi}{4}
Reply 2524
Original post by Khallil
.

Beat me to it aha
Original post by Flauta
Beat me to it aha


It was a 1* question. How could I not jump at the chance?! :tongue:
Reply 2526
Original post by Khallil
It was a 1* question. How could I not jump at the chance?! :tongue:


Indeed. I'm hoping that someone else will post an unnecessarily complicated solution now :smile:
(edited 10 years ago)
Reply 2527
Hurrah, I've reignited this thread!

Problem 214**

Find

01x21log(x)dx.\displaystyle \int_0^1 \frac{x^2-1}{\log(x)}dx.
Original post by henpen
Hurrah, I've reignited this thread!

Problem 214**

Find

01x21log(x)dx.\displaystyle \int_0^1 \frac{x^2-1}{\log(x)}dx.


log3
Reply 2529
Original post by henpen
Hurrah, I've reignited this thread!

Problem 214**

Find

01x21log(x)dx.\displaystyle \int_0^1 \frac{x^2-1}{\log(x)}dx.



Original post by Tarquin Digby
log3


Any hints?
Original post by CD315
Any hints?


Differentiate 01xa1logxdx\displaystyle \int_0^1 \frac{x^a-1}{\log{x}}\,\mathrm{d}x w.r.t a.
(edited 10 years ago)
Reply 2531
Problem 216**

Find

0dxacos2(x)+bsin2(x). \displaystyle \int_0^\infty \frac{dx}{a\cos^2(x)+b\sin^2(x)}.

Problem 217**

Find

0πlog(12αcos(x)+α2)dx,αR,α1. \displaystyle \int_0^\pi \log(1-2\alpha \cos(x)+\alpha^2)dx, \alpha \in \mathbb{R}, |\alpha|\ne 1.
(edited 10 years ago)
Problem 218*

Evaluate
ππsin(nx)(1+2x)sinxdx.\displaystyle \int_{-\pi}^{\pi} \frac{sin(nx)}{(1+2^x)sinx}dx.
For natural n.

A word of advice: * doesn't mean it's easy...
Original post by henpen
Problem 216**

Find

0dxacos2(x)+bsin2(x). \displaystyle \int_0^\infty \frac{dx}{a\cos^2(x)+b\sin^2(x)}.

Problem 217**

Find

0πlog(12αcos(x)+α2)dx,αR,α1. \displaystyle \int_0^\pi \log(1-2\alpha \cos(x)+\alpha^2)dx, \alpha \in \mathbb{R}, |\alpha|\ne 1.


216....

(ab)^-0.5*arctan(pi/2*sqrt(b/a))

?
Original post by Llewellyn
Problem 218*

Evaluate
ππsin(nx)(1+2x)sinxdx.\displaystyle \int_{-\pi}^{\pi} \frac{sin(nx)}{(1+2^x)sinx}dx.
For natural n.

A word of advice: * doesn't mean it's easy...


u=-x, and im not suggesting its an odd function

in thi case * did mean easy
Original post by Tarquin Digby
I thought you're not allowed to integrate over a discontinuity? (denominator is 0 when x=0)


u can in certain cases, you have to see what it comes out as and then youll know
Original post by henpen
Problem 413*

Find

0π211+tanα(x)dx.\displaystyle \int_0^\frac{\pi}{2} \frac{1}{1+\tan^\alpha(x)}dx.


funny how everyone was astounded when LOTF got this right at the beginning of the thread (a specific case so slightly easier) but now everyones calling it an easy question

:colone:
Original post by Llewellyn
Problem 218*

Evaluate
ππsin(nx)(1+2x)sinxdx.\displaystyle \int_{-\pi}^{\pi} \frac{sin(nx)}{(1+2^x)sinx}dx.
For natural n.

A word of advice: * doesn't mean it's easy...


Solution 218:

Spoiler



Original post by Tarquin Digby
I thought you're not allowed to integrate over a discontinuity? (denominator is 0 when x=0)


In this case you can because the function is well behaved as you approach that particular discontinuity.
(edited 10 years ago)
Original post by henpen
Problem 413*

Find

0π211+tanα(x)dx.\displaystyle \int_0^\frac{\pi}{2} \frac{1}{1+\tan^\alpha(x)}dx.


funny how everyone was astounded when LOTF got this right at the beginning of the thread (a specific case so slightly easier) but now everyones calling it an easy question

:colone:
Original post by Llewellyn
Problem 218*

Evaluate
ππsin(nx)(1+2x)sinxdx.\displaystyle \int_{-\pi}^{\pi} \frac{sin(nx)}{(1+2^x)sinx}dx.
For natural n.

A word of advice: * doesn't mean it's easy...


pi/0 for o/e
Original post by DJMayes
Solution 218:

Spoiler





In this case you can because the function is well behaved as you approach that particular discontinuity.


you used my hint well young padawan...maybe you should aim to apply to warwick :biggrin:

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