Question

Hey @ganeshie8! I have another (hopefully quicker) question. Can you check this:
ORIGINAL: "(x^3) - (1/(x^3))"

MY ANT.DRVTV. = ((x^4)/4) + (1/(2x^2)) + C

(?) :)

Answer 1

Correct ! good job !!

Answer 2

YAYAYAY!

Answer 3

Also for verifying the answer, u may use the wolfram :
http://www.wolframalpha.com/input/?i=%5Cint+%28x%5E3%29+-+%281%2F%28x%5E3%29%29

its a good site for checking answers :)

Answer 4

Lols, I get the hint ;) Alrighty, I'll check it out.

Answer 5

Taking the derivative of your answer and comparing it with what you started with is not bad, either, and gives you extra practice doing derivatives :-)

Answer 6

thats a very good idea actually, instead of dumbly checking in wolfram :)

Answer 7

take the derivative of ur answer,
u should get back the function u started wid

Answer 8

there are cases where you may end up with something that is equivalent but expressed differently, say in trig stuff, but then go ahead and use wolfram alpha :-)

Answer 9

Ahhhhh... That's awesome! I didn't put two and two together until you mentioned that... That's brilliance. :))

Answer 10

Haha, okay. Good deal :)

Answer 11

Wolfram is the lazy way. :P

Answer 12

but it's like multiplying out the result to check your factoring work: it helps you recognize the patterns so the next time you might not even have to do the work, or as much of it.

Answer 13

Lols. Hating a bit are we ;)

Answer 14

Thank you to all who positively contributed! You were immensely helpful. :)

Answer 15

I'm not hating, I run Mathematica (desktop version of Wolfram Alpha, you might say) all the time. But I try to do enough of the work by hand that I keep the skills sharp, and use the big M to keep me honest or the real drudge work. I'm all for having computers do the drudge work, as long as you're still able to do it yourself!

Answer 16

Lols, sorry. I was talking to Isaiah, haha. :) sorry, sorry, sorry!!

Answer 17

I didn't hit enter before you posted under him.

Answer 18

hahah happens xD 'lazy' is very much a positive word... u knw there is a theory that says somthing like : all efficient folks are lazy...

Answer 19

“Laziness is the first step towards efficiency.”
― Patrick Bennett

u wil get to hear these kindof stuff more if u ever enter into computer science careers...

Answer 20

lol jk ;)

Answer 21

Lols, nice.... ;)

Answer 22

or as GB Shaw said, "The reasonable man adapts himself to the world. The unreasonable man persists in trying to adapt the world to himself. All progress, therefore, depends upon the unreasonable man."

Answer 23

lol nice one xD all inventors/scientists are unreasonables :P

Answer 24

Oh God... Turned the page: mind-stoppage. Okay, so my next one is:
"piCOSpiX"

Answer 25

Ick, that looks bad. Umm,
Here: "(pi)(cos(pi)x)" ...O believe is how that would be broken up (?)

Answer 26

*I believe...

Answer 27

heard of 'substitution method ' before ?

Answer 28

\(\large \int \pi \cos(\pi x) dx\)

Answer 29

Is it related to the Chain Rule?

Answer 30

yup !!

Answer 31

or u can use advanced guessing if u want

Answer 32

I haven't seen that before, but okay, I'm familiar with the Chain Rule.

Answer 33

. . . So, I'm assuming I should use form above. What's the first step?

Answer 34

substitute,
\(\pi x = t \)

differentiate both sides :
\(\pi dx = dt \)

solve for \(dx\)
\(dx = \frac{dt}{\pi}\)

Answer 35

so the integral becomes :
\(\large \int \pi \cos(t) \frac{ dt}{\pi} \)

\(\large \int \cos(t) dt \)

Answer 36

Where did the "t", and the "dt/pi" come from... (?)

Answer 37

integral of \(cos(t)\) is simply \(\sin(t) + c\)
plug back the t value, the final answer wud be :
\(\sin(\pi x) + c\)

Answer 38

thats one way, its called 'substitution method'

Answer 39

So... "t" represents whatever 'value' is in place of the theta value, representing it as a term?

Answer 40

yes, we substituted \(\pi x = t\)

Answer 41

So... What happens to the first pi, before the cos?

Answer 42

good question :) pi is just a constant. so it stays same.

Answer 43

So... To find the anti derivative of this, we take the derivative of the whole thing
[ =(pi(sin(pix)) ] Times d(theta), or dt ?
So... Is it:
pi(SIN(pi))

Answer 44

Whoops, "+ C"

Answer 45

? Am I even remotely on the right track? :/

Answer 46

yes you're on right track, but lets do it from scratch again.. to get u use to 'substitution method' :)

Answer 47

Okay.

Answer 48

first step : substitute \(\pi x = t\), and find \(dx\) in terms of \(dt\)

Answer 49

\(\pi x =t \)

differentiate both sides, wat do u get ?