Question

How can i solve integral of negative exponents?? like u^-2.. ? Thank you..

Answer 1

power rule works for both negative and positive exponents (-1 is the only exception)

Answer 2

\[\large \int x^n dx = \dfrac{x^{n+1}}{n+1} + C\]

Answer 3

What if the exponent is -1??

Answer 4

haha then everything falls apart and the world comes to end

Answer 5

\[\large \int x^{-1}dx = \int \dfrac{1}{x}dx = \ln |x| + C\]

Answer 6

\[ \large \int {1 \over x} dx = ln |x| \]

Answer 7

Ok. Thank you for the answers!! :) :) :)

Answer 8

Can you give me techniques on how to analyze equations to solve b4 solving them? I'm currently taking up differential equations and I'm having difficulties in integrals..

Answer 9

The one I alway get wrong:

\[ \large \int k dx = kx \]

I alway try to differentiate the constant instead of integrating.

Answer 10

familiar with separation of variables ?

Answer 11

Yes. But after separating variables, here comes integration and as you said earlier, the world comes to an end..

Answer 12

Oh yes, integration is a different story - to my knowledge, the most difficult part is solving the DE... not integration

Answer 13

solving the DE is same as isolating the dependent variable "y"

Answer 14

usually, in a DE course, you will be given equations such that they turn out into simpler integrals... the main focus is on solving the DE, not on integration.

Answer 15

knowing below integral tricks are sufficient most of the time :
1) u substitution
2) trig substitution
3) partial fractions
`4) integration by parts ` (no need to master this)

Answer 16

OoooKKK. Guess I'll have to review those topics overnight.. Thank you so so much.. :)

Answer 17

There is also what Prof, Jerison calls "advanced guessing"

\[ \large \int xe^{-x^2} dx \]

You can guess \( e^{-x^2} \) and then differentiate to see what needs adjusting.

In this case

\[\large {d \over {dx}} e^{-x^2} = -2xe^{-x^2}\]

and \(-{! \over 2} \) of that is our integrand. So

\[ \large \int xe^{-x^2} dx = - {1 \over 2} e^{-x^2} \]

Answer 18

you're good to start DE course if you know how to take below integral using u substitution :
\[\large \int 2xe^{x^2}~ dx\]

Answer 19

mmm

Answer 20

you took Linear algebra before ? @kai021

Answer 21

Advance algebra, it is.. 1st semester. I think my biggest problem is recalling what previous lessons..

Answer 22

Don't wry, DE is not that hard... you can review everything parallel

Answer 23

how linear is related to this ?

Answer 24

solving systems of equations, matrix exponentials, matrix methods for inhomogeneous systems... all these require some background in LA

Answer 25

try. :) i let \[u=e ^{x ^{2}}\]
and du=\[=xe ^{x ^{2}}\]dx

Answer 26

that covers almost 1/4th of any DE course

Answer 27

xD

Answer 28

@ikram002p
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-29-matrix-exponentials/

Answer 29

Wow ! thats a very unusual try !!! but it works xD
it never occurred to me before !

Answer 30

\[\large \int 2xe^{x^2}~ dx \]

Answer 31

so, you want to let \(\large u = e^{x^2}\) ?
\(\large \implies du = e^{x^2} (2x) dx \)
\(\large \implies du = 2xe^{x^2}dx \)

So, the integral becomes :

\[\large \int du\]

Answer 32

Wait. my approach is wrong.. :3

Answer 33

which is same as :


\[\large \int 1 du\]

\[\large u + C\]

\[\large e^{x^2} + C\]

Answer 34

nope, your approach is perfectly correct ! and i must say its the best substitution !

Answer 35

Should it be u=e^(x^2) and v= x ???

Answer 36

avoid integration by parts when u can

Answer 37

always try substitution first; try integration by parts only after exhausting all other options.

Answer 38

Oh. ok. :) Hope I won't get stuck with integrating tomorrow. We'll have our preliminary exams.. :) Thank you!! :)

Answer 39

good luck !

Answer 40

:)

Answer 41

A fairly good way to reason out why x^-1 doesn't work like all the rest is because if you differentiate \[\Large \frac{d}{dx}(x^0)=0*x^{-1}\]

Of course, anything to the zero power is a constant, so that's why you get zero here. The problem is we really don't have any standard way to get there haha.

Answer 42

ok lol within week ill master DEs :P
both ordinary and partial xD