Integration by completing the square

In summary, "integration by completing the square" is a method used in calculus to solve integrals involving quadratic expressions. It is typically used when the integrand is a quadratic expression or contains a combination of quadratic and linear terms. The method involves transforming the expression into a perfect square form, allowing for the use of the substitution method. The steps for using "integration by completing the square" include rewriting the integrand, adding and subtracting a constant, using substitution, and then replacing the substituted variable. This method has various applications in fields such as physics, engineering, economics, and finance.
  • #1
luigihs
86
0
1/x^2 + 4x + 5



1) Completing the square
x^2 + 4x/2 + 5 - 4
(x+2)^2 +1

After this I know what to do??
 
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  • #2
luigihs said:
1/x^2 + 4x + 5
1) Completing the square
x^2 + 4x/2 + 5 - 4
(x+2)^2 +1

After this I know what to do??
What's your question? You're asking about an integration problem - what is the problem?

x2 + 4x + 5 = x2 + 4x + 4 + 1 = (x + 2)2 + 1
 
  • #3
What you wrote is
[tex]\frac{1}{x^2}+ 4x+ 5[/tex]
but I feel sure you meant
[tex]\frac{1}{x^2+ 4x+ 5}= \frac{1}{(x+2)^2+ 1}[/tex]

Now, do you know an integral formula for
[tex]\int \frac{dx}{x^2+a^2}[/tex]?
 
  • #4
I misread that 1/ part as "problem #1". Didn't occur to me that he meant the reciprocal of something.
 

Related to Integration by completing the square

What is "integration by completing the square"?

"Integration by completing the square" is a method used in calculus to solve integrals that involve quadratic expressions. It involves transforming the quadratic expression into a perfect square form, making it easier to integrate.

When is "integration by completing the square" used?

"Integration by completing the square" is typically used when the integrand (the expression being integrated) is a quadratic expression. It is also useful when the integrand contains a combination of quadratic and linear terms.

How does "integration by completing the square" work?

The method of "integration by completing the square" involves adding and subtracting a constant term to the integrand in order to create a perfect square expression. This then allows for the use of the substitution method to solve the integral.

What are the steps for using "integration by completing the square"?

The steps for using "integration by completing the square" are as follows:

  • 1. Rewrite the integrand in the form of a perfect square expression.
  • 2. Add and subtract a constant term to the integrand, as necessary.
  • 3. Use the substitution method to solve the integral.
  • 4. Replace the substituted variable with its original expression to get the final solution.

What are some common applications of "integration by completing the square"?

"Integration by completing the square" is commonly used in physics and engineering to solve problems involving quadratic equations, such as finding the area under a parabolic curve or calculating the displacement of an object under constant acceleration. It can also be used in economics and finance to calculate the present value of future payments.

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