How would you approach this integral?

In summary, there are several methods for solving integrals, such as substitution, integration by parts, trigonometric substitution, and partial fractions. The first step in approaching an integral is to simplify the integrand as much as possible. Improper integrals can be solved by breaking them into smaller, finite parts and using limits as needed. U-substitution involves substituting a variable for a more complicated expression within the integrand. To check the correctness of a solution, you can differentiate the antiderivative, graph the integrand and antiderivative, or use numerical methods for approximation.
  • #1
Lorena_Santoro
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  • #2
$u=x-3$
 
  • #3
Let u= x- 3. Then x= u+ 3 and dx= du so the given integral is equal to $\int (u+ 3)u^{1/2}du= \int u^{3/2}+ 3u^{1/2}= \frac{2}{5}u^{5/2}+ 2u^{3/2}+ C= \frac{2}{5}(x- 3)^{5/2}+ 2(x- 3)^{3/2}+ C$.
 

FAQ: How would you approach this integral?

How do you determine the limits of integration for an integral?

The limits of integration for an integral can be determined by looking at the given function and identifying the points where the function changes or crosses the x-axis. These points will serve as the lower and upper limits of integration.

What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration and will result in a numerical value. An indefinite integral has no limits and will result in a function with a constant of integration.

How do you choose the appropriate integration technique for a given integral?

The appropriate integration technique can be determined by looking at the form of the integrand. Some common techniques include substitution, integration by parts, and partial fractions.

Can integrals be solved without using calculus?

In some cases, integrals can be solved without using calculus by using geometric or algebraic methods. However, calculus is typically the most efficient and accurate method for solving integrals.

What are some real-world applications of integrals?

Integrals have many real-world applications, including calculating areas and volumes, determining displacement and velocity, and finding the average value of a function. They are also used in physics, engineering, economics, and other fields.

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