Integral Help: Solving $\int \frac{1}{(a+x^2)\sqrt{2a+x^2}}\,dx$

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In summary, the concept of "Integral Help" involves finding the definite or indefinite integral of a given function. This involves using various methods, such as substitution, to solve integrals. The constant "a" in the integral represents the coefficient of the $x^2$ term and can affect the difficulty of solving the integral. Integrals have many real-life applications, such as calculating work, determining motion and probabilities, and solving engineering and physics problems. While there are multiple methods for solving integrals, substitution is the most straightforward for the example integral provided.
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Identity
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It's been a while since I've done much integration, could someone please give me a hint with:

[tex]\int \frac{1}{(a+x^2)\sqrt{2a+x^2}}\,dx[/tex]

Thanks
 
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  • #2
Use Euler substitution:

[tex]
\sqrt{2a + x^{2}} = x - t
[/tex]
 
  • #3
Thanks :)
 

1. What is the basic concept of "Integral Help"?

The concept of "Integral Help" involves finding the definite or indefinite integral of a given function. It is a fundamental concept in calculus that helps in determining the area under a curve, the volume of a solid, and many other important calculations in mathematics, physics, and engineering.

2. How do you solve the integral $\int \frac{1}{(a+x^2)\sqrt{2a+x^2}}\,dx$?

To solve this integral, you can use a substitution method. Let $u = 2a + x^2$, then $du = 2xdx$. Substituting these values into the integral, we get: $\int \frac{1}{u\sqrt{u}}\frac{du}{2}$. Simplifying this, we get: $\frac{1}{2}\int u^{-\frac{3}{2}}\,du = -\frac{1}{\sqrt{u}} + C$. Substituting back for $u$, we get the final answer: $-\frac{1}{\sqrt{2a + x^2}} + C$.

3. What is the significance of the constant "a" in the integral $\int \frac{1}{(a+x^2)\sqrt{2a+x^2}}\,dx$?

The constant "a" in the integral represents the coefficient of the $x^2$ term. It determines the shape and position of the graph of the function, and can affect the difficulty of solving the integral. Different values of "a" may require different methods of integration.

4. Can the integral $\int \frac{1}{(a+x^2)\sqrt{2a+x^2}}\,dx$ be solved using other methods besides substitution?

Yes, there are other methods that can be used to solve this integral, such as partial fractions or trigonometric substitution. However, substitution is the most straightforward method for this particular integral.

5. What are some real-life applications of solving integrals?

Integrals have various real-life applications, such as calculating the work done by a force, finding the center of mass of an object, determining the velocity and acceleration of an object, and calculating probabilities in statistics. They are also used in engineering and physics to solve problems related to motion, heat transfer, and fluid dynamics, among others.

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