El_Arrow
  • El_Arrow
please need help with these two integral problems
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
El_Arrow
  • El_Arrow
\[\int\limits_{0}^{\pi/2} \sin^2x*\cos^2x\]
El_Arrow
  • El_Arrow
and \[\int\limits_{?}^{?} \sin8x * \cos5x\]
El_Arrow
  • El_Arrow
@Concentrationalizing @perl @zepdrix @freckles

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El_Arrow
  • El_Arrow
how do i do them?
anonymous
  • anonymous
in the first one just multiply and divide by 4 make it sin^2 2x then convert it using 1-cos 2x = 2sin^2 x it will become a basic integral
El_Arrow
  • El_Arrow
why by 4?
anonymous
  • anonymous
and for the second one 2Sin A cos B = Sin (A+B) + Sin (A-B)
freckles
  • freckles
\[\sin(x+y)=\sin(x)\cos(y)+\sin(y) \cos(x) \\ \sin(x-y)=\sin(x)\cos(y)-\sin(y)\cos(x) \\ \text{ add these identities } \sin(x+y)+\sin(x-y)=2 \sin(x) \cos(y) +0 \\ \text{ now divide by } 2 \\ \frac{\sin(x+y)+\sin(x-y)}{2}=\sin(x) \cos(y) \\ \frac{1}{2} \sin(x+y)+\frac{1}{2} \sin(x-y)=\sin(x)\cos(y)\] for the second one (Which I'm late for )
anonymous
  • anonymous
coz sin 2x = 2 sin x cos x and sin ^2 2x = 4sin^2 x cos^2 x this will replace those two terms
El_Arrow
  • El_Arrow
okay so i got sin(13)+sin(3) for the second one
El_Arrow
  • El_Arrow
so now i divide the whole thing by 2?
anonymous
  • anonymous
check this exp again sin (13) + sin (3) isn't there something missing?
El_Arrow
  • El_Arrow
oh the 1/2's right
anonymous
  • anonymous
its 'x' dear
anonymous
  • anonymous
sin (13x) + sin (3x)
El_Arrow
  • El_Arrow
x's where do they go?
anonymous
  • anonymous
got it now?
El_Arrow
  • El_Arrow
so thats the final answer?
anonymous
  • anonymous
no, now you have to integrate them wrt x
El_Arrow
  • El_Arrow
how?
El_Arrow
  • El_Arrow
oh you mean turn them to cosines
anonymous
  • anonymous
yeah
El_Arrow
  • El_Arrow
okay so it would be 13cos(13x) + 3cos(3x)
El_Arrow
  • El_Arrow
right?
anonymous
  • anonymous
you should see your notes again those 12 and 3 will be in the denom
El_Arrow
  • El_Arrow
okay fixed it so for the first one its a trig identity right?
anonymous
  • anonymous
yeah
El_Arrow
  • El_Arrow
so when its sin^2x*cos^2x you always change it to 4sin^2x*cos^2x?
anonymous
  • anonymous
yep
El_Arrow
  • El_Arrow
okay
El_Arrow
  • El_Arrow
okay so i got |dw:1433738264454:dw|
El_Arrow
  • El_Arrow
whats the next step @divu.mkr
El_Arrow
  • El_Arrow
@freckles do you know how to do the first one?
freckles
  • freckles
\[\int\limits _0^\frac{\pi}{2}\sin^2(x) \cos^2(x) dx \\ \int\limits_0^\frac{\pi}{2} \frac{1}{4} 4 \sin^2(x) \cos^2(x) dx \\ \frac{1}{4} \int\limits_0^\frac{\pi}{2} (2 \sin(x) \cos(x))^2 dx \\ \frac{1}{4} \int\limits_0^\frac{1}{2}( \sin(2x))^2 dx \\ \] there is an awesome idendity here you should remember for this sin things with even powers or cos with even power \[\sin^2(u)=\frac{1}{2}(1-\cos(2u)) \\ \text{ and the other one \to remember } \\ \cos^2(u)=\frac{1}{2}(1+\cos(2u))\]
freckles
  • freckles
\[\frac{1}{4}\int\limits_0^\frac{1}{2} \frac{1}{2}(1-\cos(2[2x])) dx\]
El_Arrow
  • El_Arrow
my god its a lot to memorize
freckles
  • freckles
I won't lie remembering a lot of formulas actually does kind of help :p and cuts down on time to derive other formulas(identities)
El_Arrow
  • El_Arrow
so when did you use u-substitution?
freckles
  • freckles
I didn't yet
El_Arrow
  • El_Arrow
cause you changed the pi/2 to 1/2
freckles
  • freckles
you could those on the 4x thing inside the cos thing
freckles
  • freckles
oh that was a type-o
El_Arrow
  • El_Arrow
oh okay so how many trig identities should i know
El_Arrow
  • El_Arrow
i mean just identities not trig
freckles
  • freckles
that is hard to say http://www.purplemath.com/modules/idents.htm this page has a lot of useful trig identities other identities ... well I remember pascal's triangle to help expand binomials to some positive power
freckles
  • freckles
there is also methods that are useful to depending on what you are doing
freckles
  • freckles
like you pretty much want to remember everything you can from your algebra and trig days
freckles
  • freckles
calculus pretty much depends on all of that
El_Arrow
  • El_Arrow
yeah this is why i hate calculus
freckles
  • freckles
@ganeshie8 I think I seen a post you made recently about people hating math because of something or whatever is there anything you can say here to inspire @El_Arrow into loving calculus ? :p
freckles
  • freckles
Inspirational words needed
El_Arrow
  • El_Arrow
i just hated it cause its stressful sometimes you know
El_Arrow
  • El_Arrow
but i do enjoy working out problems and stuff
freckles
  • freckles
maybe it is just overwhelming I guess
freckles
  • freckles
you get to calculus and maybe forget a lot so you are trying to learn the calculus like things and trying to relearn everything from algebra to trig
ganeshie8
  • ganeshie8
I think calculus1 is simple and easy, it deals mainly with two problems : 1) slope of tangent line 2) area under curve Most of the times, it is actually the algebra/trig/geometry part (which is not calculus) that is painful and makes calculus look messy...
ganeshie8
  • ganeshie8
guess it just takes some time to decouple calculus from algebra and see its beauty
El_Arrow
  • El_Arrow
i remember some stuff from high school math like the trig identities and pythagoeron theorem and other stuff
El_Arrow
  • El_Arrow
yeah i guess your right just takes some time you know
freckles
  • freckles
I think I remember learning some algebra while in calculus. I think I was also still learning algebra even after finishing calculus 1. Algebra is pretty crazy stuff.
freckles
  • freckles
For trig, I think I remember a few trig identities and derive the rest from what I knew when I needed those identities
freckles
  • freckles
eventually though but long after calculus 1 I was able to recall more trig identities
ganeshie8
  • ganeshie8
for calculus1 you only need to memorize below trig identities 1) double angle identities (sin(2x), cos(2x) and tan(2x)) 2) angle sum identities (sin(a+b), cos(a+b), tan(a+b)) apart from the well known pythagorean identity sin^2x+cos^2x=1
El_Arrow
  • El_Arrow
i think that if i had taken calculus in high school i wouldnt be struggling so much
El_Arrow
  • El_Arrow
what about for calculus 2 and 3?
ganeshie8
  • ganeshie8
you wont be struggling as much if you memorize those few trig identities
El_Arrow
  • El_Arrow
what about in differential equations?
ganeshie8
  • ganeshie8
i bet you felt differentiation part of calc1 easy because you know how to differentiat almost anything using product rule, quotient rule and chain rule. Integration requires "guessing", so it confuses everyone in the start... but you will get used to it after working few problems
El_Arrow
  • El_Arrow
i hope so
ganeshie8
  • ganeshie8
`double angle` and `angle sum` are the only trig identities you will be using in most of your calc1,2,3 problems... the focus is about learning calculus, not mastering trigonometry... so professors/authors avoid problems that deal with complicated trig identities in academic courses
El_Arrow
  • El_Arrow
oh okay thank you professors/authors
El_Arrow
  • El_Arrow
well its been nice chatting with you guys but i get wake up early tomorrow for calculus class
El_Arrow
  • El_Arrow
good night and thanks for the advise
freckles
  • freckles
@ganeshie8 is so awesome that way
ganeshie8
  • ganeshie8
gnite have horrible calc dreams ;p
freckles
  • freckles
i used to have horrible math dreams
freckles
  • freckles
for real
El_Arrow
  • El_Arrow
lol thanks
freckles
  • freckles
Like I would be dreaming about trying to solve a math problem and I would be unable to do it in my dream. I would do number crunching in my sleep. Numbers everywhere in my dreams. Math can be pretty haunting.
ganeshie8
  • ganeshie8
lol that sounds real scary... of course there is a slight possibility that you solve one of those open hard problems in sleep and remember after waking up ;) this reminds me of benzene structure discovery :P
freckles
  • freckles
@ganeshie8 It was never relaxing for me to sleep back then.
freckles
  • freckles
I couldn't get the math out of my head.
ganeshie8
  • ganeshie8
there is some theory that you relax more when you dream
freckles
  • freckles
It is all lies.
ganeshie8
  • ganeshie8
something like... you dream only when you're in deep stage of sleep
freckles
  • freckles
I wonder if I was really sleep all of those times then.

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