Direct integration by substitution

In summary, direct integration by substitution is a mathematical method used to evaluate integrals by substituting a part of the integral with a new variable. It is used when the integrand is a composite function or contains a function and its derivative. To perform this method, identify the inner function, find its derivative, substitute them into the integral, solve with respect to the new variable, and then substitute the original function back in. The purpose of this method is to simplify difficult integrals. Some common mistakes to avoid include forgetting to substitute back in, choosing the wrong variable for u, not taking the derivative of u, and setting limits incorrectly.
  • #1
GeoMike
67
0
Definite integration by substitution

I just need a check on this, the book and I are getting different answers...

The problem and my answer:
http://www.mcschell.com/p14.gif

http://www.mcschell.com/p14_worked.jpg

The book gives 0.00448438 though. :confused:

Thanks!
-GeoMike-
 
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  • #2
Your answer looks right. Maybe you copied the question wrong?
 
  • #3
Your answer is correct, assuming that you gave us the correct problem.

Nice work, by the way - very neat handwriting!
 
  • #4
I'm sure they mistyped the answer in the book.

Daniel.
 
Last edited:

What is direct integration by substitution?

Direct integration by substitution is a mathematical method used to evaluate integrals, or find the area under a curve. It involves substituting a part of the integral with a new variable, making the integral easier to solve.

When is direct integration by substitution used?

Direct integration by substitution is used when the integrand (the function inside the integral) is a composite function, meaning it is made up of one function inside another. It is also used when the integrand contains a function and its derivative.

How do you perform direct integration by substitution?

To perform direct integration by substitution, follow these steps:

  1. Identify the inner function, and let u be equal to that function.
  2. Find the derivative of u, du.
  3. Substitute u and du into the integral, replacing the inner function with u and the derivative with du.
  4. Solve the new integral with respect to u.
  5. Substitute the original function back in for u.

What is the purpose of direct integration by substitution?

The purpose of direct integration by substitution is to simplify integrals that are difficult or impossible to solve using other methods. It allows us to transform a complex integral into a simpler one that can be evaluated more easily.

What are some common mistakes to avoid when using direct integration by substitution?

Some common mistakes to avoid when using direct integration by substitution include:

  • Forgetting to substitute back in for u at the end.
  • Not choosing the correct variable for u.
  • Forgetting to take the derivative of u.
  • Not setting limits correctly when evaluating definite integrals.

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