Definite integral

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[astral hedge]

what have you tried?

[astral hedge]

i expanded sin 2x

took out 2 from root 4

then took out cos^2 from root

and put tan x =u

then U sub

[astral hedge]

what

[astral hedge]

wat

i mean i solved it im getting -4/5

i wanna confirm it

[hushed stone]

Let's use the substitution:

u = √(2sin2x)
du = (2cos2x / √(2sin2x)) dx = (cos2x / u) dx

[astral hedge]

,w integrate 1/(cos^3(x)sqrt(2sin(2x))) from 0 to pi/4

[hushed stone]

Notice that we need to express everything in terms of 'u'. We can do this using the double angle formula and the Pythagorean identity

[astral hedge]

wait

ehh its definite

[hushed stone]

Notice that we need to express everything in terms of 'u'. We can do this using the double angle formula and the Pythagorean identity

[trim dust]

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[astral hedge]

😦

ok

what did i do wrong

!!

[hushed stone]

Let's use the substitution:

u = √(2sin2x)
du = (2cos2x / √(2sin2x)) dx = (cos2x / u) dx

[astral hedge]

i’m not entirely certain where you’re getting a cos^2 from

[hushed stone]

you lising

[astral hedge]

not really, no

[astral hedge]

when u expand it u get sinx cos x right

[astral hedge]

yes

so i take out cos^2 x

to get tan x

[hushed stone]

you ready im going to repeat

[astral hedge]

inside

[hushed stone]

Let's use the substitution:

u = √(2sin2x)
du = (2cos2x / √(2sin2x)) dx = (cos2x / u) dx

Notice that we need to express everything in terms of 'u'. We can do this using the double angle formula and the Pythagorean identity

[astral hedge]

oh

[hushed stone]

sin2x = u²/2
cos2x = 1 - sin2x = 1 - u²/2

Now, let's find the new limits of integration

When x = 0, u = √(2sin(20)) = 0
When x = π/4, u = √(2sin(2π/4)) = √2

[astral hedge]

there is no way this is gonna work

[hushed stone]

Substitute everything back into the original integral:

[astral hedge]

so currently you have 1/(2cos^4*rt(tan))

[hushed stone]

∫(from 0 to π/4) dx / (cos³x √(2sin2x))

= ∫(from 0 to √2) (u du) / ((1 - u²/2) * u)

= ∫(from 0 to √2) 2 du / (2 - u²)

[astral hedge]

yes yes yes

[hushed stone]

This integral can be solved using the arctangent integral formula:

[astral hedge]

then convert cos to sec

[hushed stone]

∫ du / (a² - u²) = (1/a) arctan(u/a) + C

[astral hedge]

also there is 1/2 in the outside

[hushed stone]

Therefore, the value of the definite integral is (√2 * π) / 4.

[astral hedge]

only two of those are used in the dx, right? what happened to the other two?

[astral hedge]

right, yes

[hushed stone]

look

Therefore, the value of the definite integral is (√2 * π) / 4.

[astral hedge]

wdym

[astral hedge]

this seems incorrect

[hushed stone]

huh

[astral hedge]

d/dx tan is sec^2?

[astral hedge]

yes

@lavish saddle

u wanna participatE?

[astral hedge]

so you have sec^2/(2rt(tan) * du/dx dx?

and convert sec^2 into 1+tan^2?

[astral hedge]

no

oh

ok what do you have

i have sec^2 whole square / root tan x dx

[hushed stone]

its correct

[astral hedge]

then 1+u^2 / root u du

[astral hedge]

no it's bs

there is no pi in the answer

[hushed stone]

lol

[astral hedge]

yeah, i put sec^2 in the du/dx

[astral hedge]

did i mention already that this answer is wrong?

also,

Do not use LLMs such as ChatGPT to ‘help’ other people in any channels, least of all #1015578016606343218 or #1020426321261756536 .

this is part of the rules

kindly desist or i will ping someone

follow-up on this?

[lavish saddle]

i was in class

[astral hedge]

so it will convert to tan^2 +1

which is u^2+1

i believe so

[lavish saddle]

okay we should at least warn cc

[astral hedge]

(u^2+1)/(2rt(u))

yes

[astral hedge]

i did verbally

not sure how much effect this will have, but they’re quiet for now

[astral hedge]

1/2u^(3/2) + 1/2u^(-1/2)?

yup

what’s the new endpoint, 1?

ok

0,1

we can take out 1/2 out of integral

1/2(u^3/2+u^-1/2)

[astral hedge]

i think this integrates to 1/5u^(5/2) + u^(1/2)

this might be where the error comes?

since i noticed you were off by exactly -2

integrating will be 2/5*u^5/2 + 2*u^1/2

[astral hedge]

was it a - sign?

[astral hedge]

but divided by 2?

yea divided by 1/2

so 2 goes up

actually i found my error , while integrating u^-1/2 it should have been u^1/2 divide by 1/2

but i did divide by -1/2

[astral hedge]

yes

[astral hedge]

.

this should give the correct answer now

[astral hedge]

it's 2/5 not 1/5

but you took the 1/2 out

and there is a 2 in multiplication with u^1/2

i left it in

oh

so u put that in

okay

yea

yeah

it sucks that

it was a fairly difficult ques

i did everything right

but made this error

i would give you near-full method marks if i were marking your answer

this is no more than a 1-point deduction

[astral hedge]

wdym

edited for clarity

i see ty @astral hedge

[finite pollen]

@timber zinc has given 1 rep to @dusk tartan

[astral hedge]

@astral hedge

im getting 2 pi

i put pi/2- x

as the property