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Proof of an integral property

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4

$begingroup$

$$int_0^1 f(x)g'(x)dx=pi$$ If $f(1)g(1)=f(0)g(0)$ then $$int_0^1 f'(x)g(x)dx= -pi$$
So I have to prove this and I have absolutely no idea how to do it. I am guessing I will have to use the fundamental theorem of calculus and it show that the rate of change is $1$ because it didn't change from $f(1)g(1)$ to $f(0)g(0)$

calculus integration definite-integrals

share|cite|improve this question

edited 9 hours ago

Eevee Trainer

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asked 9 hours ago

adam hanyadam hany

211

New contributor

adam hany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$

• 
  
  
  

  
  7
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know integration by parts?
  $endgroup$
  – Minus One-Twelfth
  9 hours ago
  
  
  

  
  
  
  

add a comment | 

4

$begingroup$

$$int_0^1 f(x)g'(x)dx=pi$$ If $f(1)g(1)=f(0)g(0)$ then $$int_0^1 f'(x)g(x)dx= -pi$$
So I have to prove this and I have absolutely no idea how to do it. I am guessing I will have to use the fundamental theorem of calculus and it show that the rate of change is $1$ because it didn't change from $f(1)g(1)$ to $f(0)g(0)$

calculus integration definite-integrals

share|cite|improve this question

edited 9 hours ago

Eevee Trainer

6,53811237

asked 9 hours ago

adam hanyadam hany

211

New contributor

adam hany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  7
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know integration by parts?
  $endgroup$
  – Minus One-Twelfth
  9 hours ago
  
  
  

  
  
  
  

add a comment | 

$begingroup$

$$int_0^1 f(x)g'(x)dx=pi$$ If $f(1)g(1)=f(0)g(0)$ then $$int_0^1 f'(x)g(x)dx= -pi$$
So I have to prove this and I have absolutely no idea how to do it. I am guessing I will have to use the fundamental theorem of calculus and it show that the rate of change is $1$ because it didn't change from $f(1)g(1)$ to $f(0)g(0)$

calculus integration definite-integrals

share|cite|improve this question

edited 9 hours ago

Eevee Trainer

6,53811237

asked 9 hours ago

adam hanyadam hany

211

New contributor

adam hany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited 9 hours ago

Eevee Trainer

6,53811237

asked 9 hours ago

adam hanyadam hany

211

New contributor

adam hany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited 9 hours ago

Eevee Trainer

6,53811237

asked 9 hours ago

adam hanyadam hany

211

New contributor

adam hany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited 9 hours ago

Eevee Trainer

6,53811237

edited 9 hours ago

Eevee Trainer

6,53811237

asked 9 hours ago

adam hanyadam hany

211

New contributor

adam hany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 9 hours ago

adam hanyadam hany

211

2 Answers
2

active

oldest

votes

5

$begingroup$

Hint:

Utilize integration by parts:

$$int f(x)g'(x)mathrm dx = f(x)g(x) - int f'(x)g(x) mathrm dx$$

If we have a definite integral, then this formula becomes

$$int_a^b f(x)g'(x)mathrm dx = f(b)g(b) - f(a)g(a) - int_a^b f'(x)g(x) mathrm dx$$

share|cite|improve this answer

edited 4 hours ago

mrtaurho

5,51051439

answered 9 hours ago

Eevee TrainerEevee Trainer

6,53811237

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  thank you so much for the answer ∫baf(x)g′(x)dx=−∫baf′(x)g(x)
  $endgroup$
  – adam hany
  9 hours ago
  

  
  
  
  

add a comment | 

3

$begingroup$

Hint:

$$dfrac{d(f(x)cdot g(x))}{dx}=?$$

Integrate both sides with respect to $x$ between $[0,1]$

share|cite|improve this answer

edited 4 hours ago

Eevee Trainer

6,53811237

answered 8 hours ago

lab bhattacharjeelab bhattacharjee

226k15157275

$endgroup$

add a comment | 

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5

$begingroup$

Hint:

Utilize integration by parts:

$$int f(x)g'(x)mathrm dx = f(x)g(x) - int f'(x)g(x) mathrm dx$$

If we have a definite integral, then this formula becomes

$$int_a^b f(x)g'(x)mathrm dx = f(b)g(b) - f(a)g(a) - int_a^b f'(x)g(x) mathrm dx$$

share|cite|improve this answer

edited 4 hours ago

mrtaurho

5,51051439

answered 9 hours ago

Eevee TrainerEevee Trainer

6,53811237

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  thank you so much for the answer ∫baf(x)g′(x)dx=−∫baf′(x)g(x)
  $endgroup$
  – adam hany
  9 hours ago
  

  
  
  
  

add a comment | 

5

$begingroup$

Hint:

Utilize integration by parts:

$$int f(x)g'(x)mathrm dx = f(x)g(x) - int f'(x)g(x) mathrm dx$$

If we have a definite integral, then this formula becomes

$$int_a^b f(x)g'(x)mathrm dx = f(b)g(b) - f(a)g(a) - int_a^b f'(x)g(x) mathrm dx$$

share|cite|improve this answer

edited 4 hours ago

mrtaurho

5,51051439

answered 9 hours ago

Eevee TrainerEevee Trainer

6,53811237

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  thank you so much for the answer ∫baf(x)g′(x)dx=−∫baf′(x)g(x)
  $endgroup$
  – adam hany
  9 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Hint:

Utilize integration by parts:

$$int f(x)g'(x)mathrm dx = f(x)g(x) - int f'(x)g(x) mathrm dx$$

If we have a definite integral, then this formula becomes

$$int_a^b f(x)g'(x)mathrm dx = f(b)g(b) - f(a)g(a) - int_a^b f'(x)g(x) mathrm dx$$

share|cite|improve this answer

edited 4 hours ago

mrtaurho

5,51051439

answered 9 hours ago

Eevee TrainerEevee Trainer

6,53811237

$endgroup$

share|cite|improve this answer

edited 4 hours ago

mrtaurho

5,51051439

answered 9 hours ago

Eevee TrainerEevee Trainer

6,53811237

edited 4 hours ago

mrtaurho

5,51051439

edited 4 hours ago

mrtaurho

5,51051439

answered 9 hours ago

Eevee TrainerEevee Trainer

6,53811237

answered 9 hours ago

Eevee TrainerEevee Trainer

6,53811237

3

$begingroup$

Hint:

$$dfrac{d(f(x)cdot g(x))}{dx}=?$$

Integrate both sides with respect to $x$ between $[0,1]$

share|cite|improve this answer

edited 4 hours ago

Eevee Trainer

6,53811237

answered 8 hours ago

lab bhattacharjeelab bhattacharjee

226k15157275

$endgroup$

add a comment | 

3

$begingroup$

Hint:

$$dfrac{d(f(x)cdot g(x))}{dx}=?$$

Integrate both sides with respect to $x$ between $[0,1]$

share|cite|improve this answer

edited 4 hours ago

Eevee Trainer

6,53811237

answered 8 hours ago

lab bhattacharjeelab bhattacharjee

226k15157275

$endgroup$

add a comment | 

$begingroup$

Hint:

$$dfrac{d(f(x)cdot g(x))}{dx}=?$$

Integrate both sides with respect to $x$ between $[0,1]$

share|cite|improve this answer

edited 4 hours ago

Eevee Trainer

6,53811237

answered 8 hours ago

lab bhattacharjeelab bhattacharjee

226k15157275

$endgroup$

share|cite|improve this answer

edited 4 hours ago

Eevee Trainer

6,53811237

answered 8 hours ago

lab bhattacharjeelab bhattacharjee

226k15157275

edited 4 hours ago

Eevee Trainer

6,53811237

edited 4 hours ago

Eevee Trainer

6,53811237

answered 8 hours ago

lab bhattacharjeelab bhattacharjee

226k15157275

answered 8 hours ago

lab bhattacharjeelab bhattacharjee

226k15157275