Difficulty to find this integral

In summary, the person was struggling with solving an integral and tried to use substitution to solve it, but made a mistake in their calculation. They were advised to clearly state their substitution and were given the correct expression for u^{\frac{3}{2}}.
  • #1
DDarthVader
51
0

Homework Statement


Hi! I'm not being able to solve this integral: [tex]\int \sqrt{2ax}\; dx[/tex] We have started with integral yesterday and I don't know much yet.



Homework Equations





The Attempt at a Solution


This is what I tried to solve the integral
[tex]\int \sqrt{2ax}\; dx = \int u^{\frac{1}{2}}\; du = \frac{2}{3}{}u^{\frac{3}{2}}=\frac{2}{3}\sqrt[3]{(2ax)^2}[/tex]
 
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  • #2
Assuming that a is a real constant,
[tex]\int \sqrt{2ax}\; dx = \int \sqrt{2a}\sqrt x\; dx= \sqrt{2a} \int \sqrt x\; dx[/tex]
Since [itex]\sqrt x[/itex] is the same as [itex]x^{\frac{1}{2}}[/itex], you can easily find its integral, and then multiply by [itex]\sqrt {2a}[/itex] to get the final answer.
 
  • #3
DDarthVader said:

The Attempt at a Solution


This is what I tried to solve the integral
[tex]\int \sqrt{2ax}\; dx = \int u^{\frac{1}{2}}\; du = \frac{2}{3}{}u^{\frac{3}{2}}=\frac{2}{3}\sqrt[3]{(2ax)^2}[/tex]
When you use a substitution, as you obviously did above, you need to write down what the substitution is. IOW, you should have u = <...> somewhere close by.

Also, u3/2 = ##\sqrt{u^3}##, not ##\sqrt[3]{u^2}##, which is what you had.
 

1. What makes finding this integral difficult?

The difficulty in finding an integral can depend on several factors such as the complexity of the function, the techniques used, and the limits of integration. Some integrals may require advanced methods or multiple steps to solve, making them more challenging to find.

2. Are there any tips or tricks for finding difficult integrals?

Yes, there are some strategies that can make finding difficult integrals easier. These include using substitution, integration by parts, and trigonometric identities. It is also helpful to simplify the integrand and break down the integral into smaller parts if possible.

3. How do I know when to stop trying to find an integral?

If you have exhausted all possible methods and cannot find an exact solution for the integral, it may be a sign to stop trying. In some cases, the integral may not have an analytic solution and can only be approximated numerically.

4. Can technology be used to find difficult integrals?

Yes, technology such as calculators and computer software can be used to find difficult integrals. However, it is important to understand the underlying concepts and techniques behind integration to properly use these tools.

5. Why is finding integrals important in science?

Integrals are used to find the area under a curve, which has many applications in science such as calculating volumes, center of mass, and work done. They also play a crucial role in solving differential equations, which are fundamental in many scientific fields.

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