please need help with these two integral problems

- El_Arrow

please need help with these two integral problems

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- El_Arrow

\[\int\limits_{0}^{\pi/2} \sin^2x*\cos^2x\]

- El_Arrow

and \[\int\limits_{?}^{?} \sin8x * \cos5x\]

- El_Arrow

@Concentrationalizing @perl @zepdrix @freckles

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## More answers

- El_Arrow

how do i do them?

- 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

why by 4?

- anonymous

and for the second one
2Sin A cos B = Sin (A+B) + Sin (A-B)

- 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

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

okay so i got sin(13)+sin(3) for the second one

- El_Arrow

so now i divide the whole thing by 2?

- anonymous

check this exp again sin (13) + sin (3)
isn't there something missing?

- El_Arrow

oh the 1/2's right

- anonymous

its 'x' dear

- anonymous

sin (13x) + sin (3x)

- El_Arrow

x's where do they go?

- anonymous

got it now?

- El_Arrow

so thats the final answer?

- anonymous

no, now you have to integrate them wrt x

- El_Arrow

how?

- El_Arrow

oh you mean turn them to cosines

- anonymous

yeah

- El_Arrow

okay so it would be 13cos(13x) + 3cos(3x)

- El_Arrow

right?

- anonymous

you should see your notes again
those 12 and 3 will be in the denom

- El_Arrow

okay fixed it so for the first one its a trig identity right?

- anonymous

yeah

- El_Arrow

so when its sin^2x*cos^2x you always change it to 4sin^2x*cos^2x?

- anonymous

yep

- El_Arrow

okay

- El_Arrow

okay so i got |dw:1433738264454:dw|

- El_Arrow

whats the next step @divu.mkr

- El_Arrow

@freckles do you know how to do the first one?

- 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

\[\frac{1}{4}\int\limits_0^\frac{1}{2} \frac{1}{2}(1-\cos(2[2x])) dx\]

- El_Arrow

my god its a lot to memorize

- 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

so when did you use u-substitution?

- freckles

I didn't yet

- El_Arrow

cause you changed the pi/2 to 1/2

- freckles

you could those on the 4x thing inside the cos thing

- freckles

oh that was a type-o

- El_Arrow

oh okay so how many trig identities should i know

- El_Arrow

i mean just identities not trig

- 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

there is also methods that are useful to depending on what you are doing

- freckles

like you pretty much want to remember everything you can from your algebra and trig days

- freckles

calculus pretty much depends on all of that

- El_Arrow

yeah this is why i hate calculus

- 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

Inspirational words needed

- El_Arrow

i just hated it cause its stressful sometimes you know

- El_Arrow

but i do enjoy working out problems and stuff

- freckles

maybe it is just overwhelming I guess

- 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

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

guess it just takes some time to decouple calculus from algebra and see its beauty

- El_Arrow

i remember some stuff from high school math like the trig identities and pythagoeron theorem and other stuff

- El_Arrow

yeah i guess your right just takes some time you know

- 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

For trig, I think I remember a few trig identities and derive the rest from what I knew when I needed those identities

- freckles

eventually though but long after calculus 1 I was able to recall more trig identities

- 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

i think that if i had taken calculus in high school i wouldnt be struggling so much

- El_Arrow

what about for calculus 2 and 3?

- ganeshie8

you wont be struggling as much if you memorize those few trig identities

- El_Arrow

what about in differential equations?

- 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

i hope so

- 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

oh okay thank you professors/authors

- El_Arrow

well its been nice chatting with you guys but i get wake up early tomorrow for calculus class

- El_Arrow

good night and thanks for the advise

- freckles

@ganeshie8 is so awesome that way

- ganeshie8

gnite have horrible calc dreams ;p

- freckles

i used to have horrible math dreams

- freckles

for real

- El_Arrow

lol thanks

- 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

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

@ganeshie8 It was never relaxing for me to sleep back then.

- freckles

I couldn't get the math out of my head.

- ganeshie8

there is some theory that you relax more when you dream

- freckles

It is all lies.

- ganeshie8

something like... you dream only when you're in deep stage of sleep

- freckles

I wonder if I was really sleep all of those times then.

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