Question

let f to be a differentiable function defined for all real x, where f(x) >=0 for all x[0,a]. If \[\int\limits_{0}^{a}f(x) dx=a\], then \[2\int\limits\limits_{0}^{5a}(f(\frac{ x }{ 5 })+3) dx\] must be equal to

Can anyone help me in this question. Don't know what to do.

Thanks.

Answer 1

Did you start by substitution?

Answer 2

because it said that f(x)=, so i did this\[2\int\limits\limits\limits_{0}^{5a}(f(\frac{ a }{ 5 })+3) dx\] and i got 10a*(a+3) but my teacher said that the answer must be linear.

Answer 3

Okay! Let me start then..
First of all, you should have commenced like substitution of x/5 as t.
Then what will your next step?
Can you write it?

Answer 4

Rahul
x is not a
May I know why you're replacing x by a ?

Answer 5

Good point!

Answer 6

And yeah you should start as suggested by sami. It gives you a nice integral to work with

Answer 7

\(\int\limits_{0}^{a}f(x) dx=a\)

This expression means the area under the graph of f(x) over the interval (0, a) is 'a'

Answer 8

ok thanks. i'll try.

Answer 9

@samigupta8 i didn't get what you meant by x/5 as t. Can you explain it a bit more.

Answer 10

You know substituition?

Answer 11

replcae dx by dt and also chamge the limits accordingly.

Answer 12

Replace*

Answer 13

\[2\int\limits\limits_{0}^{5t}(f(t)+3) dt\]
would it be this

Answer 14

Limits are 0-t.

Answer 15

Not 0-5t

Answer 16

ok. like that

\[2\int\limits\limits_{0}^{t}(f(t)+3) dt\]

Answer 17

Yep.

Answer 18

now should i integrate ?

Answer 19

Go ahead!

Answer 20

what is f(t). do i need to know that

Answer 21

Okay! Let me start wity basics then.

Answer 22

i got \[2(\int\limits\limits_{0}^{t}f(t) dt+3t)\]

Answer 23

If you have integral of f(x)dx as a then what can you say for integral of f(t)dt ?

Answer 24

t

Answer 25

Ah! No..

Answer 26

It is a.

Answer 27

sorry. i don't know then

Answer 28

Okay! You can easily get it if you replace t by x .

Answer 29

Then you get dt as dx. So we could see the integration for f(t)dt reduces to ?

Answer 30

Did you get it?

Answer 31

does it reduces to a

Answer 32

Yes obviously!

Answer 33

so now ??

Answer 34

Integrate ..

Answer 35

i got (a+3)t. is this correct

Answer 36

I got 2(a+3t)

Answer 37

oh yeah. i forgot to multiply with 2

Answer 38

You even multiplied the term a with t.
Which i didn't

Answer 39

and now what do we do ??

Answer 40

Now solve this ..
Put the limits from 0-5a

Answer 41

Don't forget that limit is for t !

Answer 42

I don't have enough time to do this question. I have to do other questions as well. so can you show me how to do this question. Thanks.

Answer 43

I have guided you how to solve this problem.
I don't know how should i go about giving you the ans.
Maybe @ganeshie8 devise a better way to help you out.
Sorry , i can't help more into this.

Answer 44

Ok. thanks for the help.

Answer 45

@ganeshie8. can you help me in this question

Answer 46

@welshfella can you help me in this question

Answer 47

I'm very rusty at this stuff I'm afraid (long time since I did it)

I think you plug in t = 0 and t=5a into 2 (a + 3t) so we get

2[ a + 3(5a) - ( a + 3(0)]

Answer 48

which simplifies to a linear function

Answer 49

sorry i cant be of more help

Answer 50

thanks for the help. i will write 30a as my answer.

Answer 51

It has to be 32a.