Finding K in Calculus: A Hint for Solving Challenging Integrals

[keewansadeq]

Homework Statement

Can anybody give me hint how to find K if F(x)= 3x+2

The integral lower part is not the same, , how to deal with his?

Homework Equations

The Attempt at a Solution

Please ,I need hint to start

 

[DrClaude]

Look at property (ii) following from the second theorem at http://www.sosmath.com/calculus/integ/integ02/integ02.html

 

[PeroK]

... or, why not just integrate what you've been given?

 

[Ray Vickson]

Homework Statement

View attachment 97741

Can anybody give me hint how to find K if F(x)= 3x+2

The integral lower part is not the same, , how to deal with his?

Homework Equations

The Attempt at a Solution

Please ,I need hint to start

What is preventing you from just computing $\int_2^x (3t + 2) \, dt$ and $\int_8^x (3t + 2) \, dt$?

 

[keewansadeq]

Thank you all,

The point guys is that , first integration is start from 2 and the other one is start from 8.
Hence if I have only the base 2, the answer will be 3x+2 =3x+2(but the problem is the second base is 8), can't figure out what I missing till now?

How can I equate them?

Thanks again

 

[DrClaude]

the answer will be 3x+2 =3x+2

That's not a valid answer for that definite integral.

 

[keewansadeq]

Thanks

Could you upload refer to examples with X is the boundary of indefinite integral not numbers?

 

[Samy_A]

When a math problem proves to be too difficult, it is a good idea to work on it in little pieces.

Why don't you do as has been suggested above: forget about the exercise itself, and just compute the following integral:
$\int_2^x (3t + 2) dt$.

What result do you get?

 

[HallsofIvy]

What, exactly, is your difficulty? Have you been able to find an anti-derivative for 3t+ 2? That is, can you find $\int 3t+ 2 dt$? What do you get when you substitute the upper and lower bounds and subtract?

 

[keewansadeq]

When a math problem proves to be too difficult, it is a good idea to work on it in little pieces.

Why don't you do as has been suggested above: forget about the exercise itself, and just compute the following integral:
$\int_2^x (3t + 2) dt$.

What result do you get?

The answer is 3x+2 (right)!

 

[Samy_A]

The answer is 3x+2 (right)!

No.
Let's go one more step back (as suggested by HallsofIvy): what is the indefinite integral $\int (3t +2) dt$?

 

[Ssnow]

Sorry, a stupid question but $F(x)=3x+2$ is a primitive of $f$ or it is the $f$ in the integral?

 

[Samy_A]

Sorry, a stupid question but $F(x)=3x+2$ is a primitive of $f$ or it is the $f$ in the integral?

That's not a stupid question, it's a good catch.

 

[Ssnow]

Ah ok because in one case one must apply directly the Fundamental calculus theorem and in the other side one must find before the primitive ... @keewansadeq you must reflect on this ...

 

[Mark44]

Sorry, a stupid question but $F(x)=3x+2$ is a primitive of $f$ or it is the $f$ in the integral?

That's not a stupid question, it's a good catch.

I noticed that as well. It could be that F is an antiderivative of f, or, as often happens, some posters mix upper and lower case for a single variable name.
@keewansadeq, did you intend f and F to represent different functions?

 

[HallsofIvy]

Sorry, a stupid question but $F(x)=3x+2$ is a primitive of $f$ or it is the $f$ in the integral?

I assumed it was the integrand because there would not be a specific constant, like 2, in the anti-derivative. If the itegrand is f(x)= 3x+ 2, then the integrand is $F(x)= (3/2)x^2+ 2x+ C$ where C is a constant that will cancel in the definite integral. If the integrand is the constant, 3, then the anti-derivative is 3x+ C, to be evaluated at 2 and x on one side, 8 and x on the other.

 

[Ssnow]

@HallsofIvy I have had the same doubt the fact is that it is possible that in the text of the schedule they choose a particular primitive of $3$ with $c=2$, I don't know because usually with $F$ denotes the primitive ...

 

[keewansadeq]

I noticed that as well. It could be that F is an antiderivative of f, or, as often happens, some posters mix upper and lower case for a single variable name.
@keewansadeq, did you intend f and F to represent different functions?

Yes I did, Capital F means anti antiderivative

 

[keewansadeq]

Actually,I am very goof in integration,but I am week in Fundamental calculus theorem, it is to me that derivative of integration will be the same function, that why I am confused.

Thanks all I appreciate

 

[keewansadeq]

No.
Let's go one more step back (as suggested by HallsofIvy): what is the indefinite integral $\int (3t +2) dt$?

This is simple

3(t^2)/2+2t

But the main problem is with Fundamental calculus theorem

 

[Samy_A]

Ok, so now we know that $\int f(t) dt = F(t) + C$.
That means, by the fundamental theorem of calculus, that $\int_a^b f(t) dt = F(b)-F(a)$.

Can you now compute $\int_2^x f(t) dt$ and $\int_8^x f(t) dt$?

 

[Ray Vickson]

This is simple

3(t^2)/2+2t

But the main problem is with Fundamental calculus theorem

Sorry, but I don't see how you can say that: in another thread (on volume computation) you did another, considerably harder example without any difficulty at all!

 

[keewansadeq]

Ok, so now we know that $\int f(t) dt = F(t) + C$.
That means, by the fundamental theorem of calculus, that $\int_a^b f(t) dt = F(b)-F(a)$.

Can you now compute $\int_2^x f(t) dt$ and $\int_8^x f(t) dt$?

Got it ,piece of cake, I just have small misunderstanding for something, and now it clear

 

[keewansadeq]

Sorry, but I don't see how you can say that: in another thread (on volume computation) you did another, considerably harder example without any difficulty at all!

Life is Hard