Matematické Fórum

stuart clark

$(1)\;\; \int \frac{x^2}{(x^2+6x+8)^2}dx$

$(2)\;\; \displaystyle \int^{\frac{\pi}{2}}_{0}\frac{\ln(1+k \sin^2 \theta)}{\sin^2 \theta}d \theta$ for $k>0$

Marian · 19. 09. 2018 11:10

↑ stuart clark:

Solution to (1) Zobrazit/skrýt!

Remark: The second problem seems to be much more interesting.

Brano · 19. 09. 2018 23:27 — Editoval Brano (19. 09. 2018 23:27)

↑ stuart clark:
for 2) I suggest substitution $x=\cot \theta$ and then per-partes

krakonoš · 20. 09. 2018 21:41

And what about use per partes. You will get cotg function in integrand. Then cotg x as cos x / sin x. And finaly substitution tg(x/2)?

Marian · 21. 09. 2018 22:08

↑ krakonoš:

With Brano's substitution one can easily relate sin^2 and x^2 with the help of a rational function. Finally, integration by parts leads to the required evaluation of the given integral.

stuart clark · 27. 09. 2018 11:48

Thanks ↑ Marian:↑ Brano:.

Matematické Fórum

#1 19. 09. 2018 08:46 — Editoval stuart clark (19. 09. 2018 08:51)

Rational Integration

#2 19. 09. 2018 11:10

Re: Rational Integration

#3 19. 09. 2018 23:27 — Editoval Brano (19. 09. 2018 23:27)

Re: Rational Integration

#4 20. 09. 2018 21:41

Re: Rational Integration

#5 21. 09. 2018 22:08

Re: Rational Integration

#6 27. 09. 2018 11:48

Re: Rational Integration

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