Question

anyone can give me simple example of integration by substitution with solution . i cant understand that topic.

Answer 1

okay, what is \[\int xdx\]

Answer 2

x^2/2 + c

Answer 3

okay, we could do that because we had the function, x, times its differential, dx
whenever we have a function times it's differential, we can integrate
now consider the following deviation on the last problem\[\int(x+1)dx\]you may be tempted to split this integral up\[\int(x+1)dx=\int xdx+\int dx=\frac12x^2+\frac12x+C\]that is one way to do it, but we can do it with substitution much quicker

What is the differential of \((x+1)\) ?

Answer 4

1, isn't ?

Answer 5

careful, 1 is the derivative
the differential is dx
here's how they work in case you forgot\[d(x)\implies dx\]\[d(x^2)\implies 2xdx\]\[d(\ln x)\implies\frac{dx}x\]

Answer 6

do the differential here is\[x+1\implies dx\]now here's the substitution:\[u=x+1\]and what is the differential of u ?\[d(u)\implies du\]so now we can substitute our integral in terms of all in terms of u (make sure that the du is correct too!)\[\int(x+1)dx=\int udu\]do you follow so far?

Answer 7

yes ,

Answer 8

so we can just integrate this then and...\[\int udu=\frac12u^2+C\]now if we put u=x+1 back into our solution we get\[\frac12(x+1)^2+C\]which you can expand to get\[\frac12x^2+\frac12x+\frac12+C\]since\[\frac12+C\]is still just a constant, we can just "absorb" the extra 1/2 into C to get\[\int udu=\frac12u^2+C=\frac12(x+1)^2+C\]and we can just leave the answer like that; no expansion neccessary.
Did you follow all that? It will get a little more involved shortly, so make sure you understand what I did there.

Answer 9

I should have said "since \(\frac12+C_1=C_2\) ..."
i.e. adding a constant to another constant is still just a constant, but that's not terribly important here)

Answer 10

i get it for that kind of example,

Answer 11

okay, so what about another variation?\[\int(2x+1)^3dx\]we could expand this cubic and split the integral, but that seems like a real pain, right?
but here's where we have to be smart about the integral and notice: "aha! dx is almost the differential of 2x+1 !"
that is your cue to try substituting u=2x+1(recognizing this takes some practice, but you'll get the hang of it ;) )
what is the differential of 2x+1 ?

Answer 12

d(u) = du

Answer 13

I am asking you for d(2x+1)

Answer 14

2

Answer 15

again, you CANNOT forget the dx part, it is crucial here (you will see why momentarily)

Answer 16

u mean 2 du

Answer 17

or in term of x ,

Answer 18

yeah be careful with what variables we are using
I'm going to use an arrow to represent taking the differential (this is how I do it on my work at least)\[u=2x+1\implies du=2dx\]do you follow my notation?

Answer 19

ok

Answer 20

I set u=2x+1 and took the differential of both sides....

Answer 21

let's look at our original integral again\[\int(2x+1)^3dx\]well, we have u=2x+1 so we can put \[\int u^3dx\] but we can't integrate with u's and x's, we need u and du, and our du=2dx, but we only have dx in the integrand! so what do we do? solve for dx...

Answer 22

since we have\[u=2x+1\implies du=2dx\]solving for dx gives\[dx=\frac12du\]now that we have an expression for dx, we can totally sub into our integral!\[\int(2x+1)^3dx=\int u^3(\frac12du)=\frac12\int u^3du\]now that's a much easier integral, and we didn't need to expand that horrible cubic.
So now we can just integrate wrt u and sub back in for x at the end...

Answer 23

\[\frac12\int u^3du=\frac18 u^4+C=\frac18(2x+1)^4+C\]a LOT easier than expanding, don't you agree?

Answer 24

any questions?

Answer 25

indeed.. what if there is 2x as the cooefficient for (2x+1)^3

Answer 26

\[\int2x(2x+1)^3dx\]here you would either have to expand or use a technique called integration by parts, which judging by the level of your questions you have not learned yet.
let me give one more difficult example though that may help you see how far this idea can be applied...

Answer 27

\[\int2x(x^2+5)dx\]I'll let you try this one on your own mostly
what should we make our u be here? any ideas?

Answer 28

excuse me, i meant to give\[\int2x(x^2+5)^4dx\]as the problem

Answer 29

pro tip: usually you want to consider what is in the parentheses for u and see how that plays out once you find its differential

Answer 30

u =x^2 + 5

Answer 31

good, so what is du ?

Answer 32

du = 2x dx
dx = 1/2x du

Answer 33

nice, but in this case we didn't have to solve for dx explicitly (though you committed no error in doing so, it makes more work for yourself than is necessary). let me show you...

Answer 34

\[\int2x(x^2+5)^4dx\]\[u=x^2+5\implies du=2xdx\]but look, if we just rearrange the terms of our integral we get\[\int(x^2+5)^4(2xdx)\]so we can just directly sub\[u=x^2+5\]\[du=2xdx\] and get\[\int(x^2+5)^4(2xdx)\]\[\int u^4du\]which should be pretty straightforward from here.
Always look out for when the differential du is already in the integral, that saves some serious work.

Answer 35

for a nice source for explanations, examples, and problems, I suggest this page (and the whole site in general)
http://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleDefinite.aspx

Answer 36

okay , i will revise back 4 better understanding , thank you very mch .

Answer 37

You're welcome :)

Answer 38

i forgot to ask , what happens to the coefficient ?