Mastering Antiderivatives: Solving Challenging Problems with U-Substitution

In summary, the conversation discusses finding the integral of a function using algebraic methods and u-substitution. The individual is preparing for a test and is seeking guidance on how to approach the problem. However, they later realize that the solution can be found by simply dividing out terms.
  • #1
dontdisturbmycircles
592
3

Homework Statement


Find [tex]\int \frac{x^{2}+3x+7}{\sqrt{x}}dx[/tex]


Homework Equations





The Attempt at a Solution




I am reviewing for a test on tuesday. I can't see a good algebraic method to solve this using the fact that [tex]\int f(g(x))g'(x)dx=F(g(x))+C[/tex] so I am starting to think it requires a u-substitution which I am trying to get better at right now before the test. Can someone point me in the right direction here? Thankyou very much, I appreciate any help!
 
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  • #2
Duh nevermind, I can just divide out the terms... *Sigh* I always spend lots of time on a problem, post it here, and then 3 minutes later I figure it out... Grr, sorry!

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Related to Mastering Antiderivatives: Solving Challenging Problems with U-Substitution

What is the concept of U-Substitution?

U-Substitution, also known as the Reverse Chain Rule, is a technique used to solve complicated integrals by substituting a function within the integral with a simpler function. This simplifies the integral and makes it easier to solve.

When should I use U-Substitution?

U-Substitution is most commonly used for integrals involving functions that are composed of other functions, such as exponential, logarithmic, and trigonometric functions. It is also useful for solving integrals with fractions or radicals.

How do I perform U-Substitution?

To perform U-Substitution, follow these steps:1. Identify the "inside" function, denoted as u.2. Take the derivative of u, du.3. Substitute u and du into the integral, replacing the "inside" function and its derivative.4. Simplify the integral using basic algebra.5. Integrate the new, simplified integral.6. Substitute the original variable back into the solution.

What are some common challenges when using U-Substitution?

Some common challenges when using U-Substitution include identifying the "inside" function, correctly taking the derivative of that function, and choosing the appropriate substitution for the integral. It may also be challenging to simplify the integral and perform the necessary algebraic manipulations.

Can U-Substitution be used for all integrals?

No, U-Substitution may not work for all integrals. It is most effective for integrals involving composite functions, but may not be useful for integrals with other types of functions, such as polynomial functions. In some cases, other integration techniques, such as integration by parts, may be more effective.

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