Integration way harder then what it looks like

[limpid sentinel]

I can’t solve it tried it with many ways. I have the answer but can’t find the solution anywhere or do it myself.

The answer is given in the spoiler so you can’t match it yourself.

[limpid sentinel]

Can* match it

[light bluff]

Did you try partial fraction decomposition?

[civic venture]

i think it is the only way

@limpid sentinel this is the way

you will have to solve this equation

by the way, which book are you using ?

[coarse skiff]

another way maybe a bit easier is to substitute x=2tan(t)

You get cos⁶t. Then express cos(t)^3 in terms of cos(3t) and square it. So, you easily get
32cos⁶t=cos(6t)+6cos(4t)+15cos(2t)+10. Integrate it. Then express it back in cos(2t), sin(2t) which then is easy in terms of tan(t)=x/2. So, yes, this should be way shorter on paper.

[ebon sage]

you can also do it by induction with $I_p = \int \frac{dx}{(4+x^2)^p}$

[scenic phoenix]

bloubbloub

[ebon sage]

also you can't decompose it further with PFD

[civic venture]

i did not thinked about it but good idea!

[civic venture]

yes or use integration by parts, there is a formula $I_n=\int_{}^{}cos^{n}(x)dx=\frac{cos^{n-1}(x)sin(x)-(n-1)\cdot I_{n-2}}{n}$

[scenic phoenix]

barann0056

[civic venture]

my bad it is :
$In=\int{}^{}cos^{n}(x)dx=\frac{cos^{n-1}(x)sin(x)+(n-1)\cdot I_{n-2}}{n}$

[scenic phoenix]

barann0056

[limpid sentinel]

I did try that cause I saw x^2+4 but it’s didn’t work it was just too lengthly and I couldn’t find a way

[limpid sentinel]

It’s given by our teacher in homework sheet I don’t know his source

Sir himself was unable to solve it in doubt class

He tried tan2x substitution and by part too but it was too lengthily and was going nowhere

[civic venture]

i mean it is pretty hard if you did not worked a lot on computation

so its okay if you don't succed this one

good luck for your calculus class

[coarse skiff]

Why, it should be no more than a page of formulas.

[coarse skiff]

It perfectly works.

[civic venture]

because you are not lazy

.close

@limpid sentinel type .close

[limpid sentinel]

.close

[silent prairie]

Solved

Post marked as solved by @limpid sentinel.

Use .unsolved if this was a mistake.

[limpid sentinel]

Yea I got it already now

Same thing I did

Still thanks

Still very lengthy took very long