Solving an Integral with Homework Help

• Melawrghk
In summary, the problem involves integrating the function \frac{26}{(169x^2+1)^2} using substitution and integration by parts. The attempt involves converting the denominator to have a square root, using an inverse substitution, and simplifying the resulting equation. However, the integration by parts step leads back to the original function, so the use of a double angle formula is suggested to solve the problem. The conversation also includes a side note about formatting equations.
Melawrghk

Homework Statement

$$\int \frac{26 dx}{{(169x^2+1)}^2}$$ <= the whole denominator is supposed to be squared...

The Attempt at a Solution

So I converted the thing in the denominator so that it has a square root:
$$\int \frac{26 dx}{{\sqrt{169x^2+1}}^4}$$

Looking at the denominator, I realized I should do an inverse substitution:
13x=tan(t)
dx=sec2(t)*dt/13

I subbed that into the equation before and got:
$$\int \frac{26 * sec^2(t) * dt}{13*sec^4(t)}$$

Simplifying which, I get:
2$$\int cos^2(t) dt$$

Then I tried doing integration by parts, but I got nowhere - I kept getting cos^2 again... Please help me, this question frustrates me. Thanks in advance!

(Sidenote: I finally got my formulas all pretty, yay!)

There's not really much profit into converting x^2 into sqrt(x)^4, is there? The rest of the general approach looks fine. You get sec^2/sec^4. So sure, cos^2(x). You just want to use a double angle formula cos(x)^2=(1+cos(2x))/2.

Thanks! I know there isn't a point with the root, I'm just more used to seeing it like that. And thanks for the double angle formula. I always forget they exist... I guess it's time I memorize them :)

What is an integral?

An integral is a mathematical concept that represents the area under a curve. It is used to find the total value of a function over a given interval.

Why is solving an integral important?

Solving an integral is important because it allows us to determine the exact value of a function over a given interval, which can be useful in many real-world applications such as calculating volumes, areas, and probabilities.

What are the different methods for solving an integral?

There are several methods for solving an integral, including the substitution method, integration by parts, and partial fractions. Each method is used to solve different types of integrals and has its own set of rules and techniques.

How can I improve my skills in solving integrals?

To improve your skills in solving integrals, it is important to practice regularly and familiarize yourself with the different methods and techniques. You can also seek help from a tutor or use online resources, such as tutorials and practice problems, to strengthen your understanding.

Can I use a calculator to solve integrals?

Yes, there are certain calculators that have built-in integral solving functions. However, it is important to understand the concepts and methods of solving integrals by hand before relying on a calculator. Also, not all integrals can be solved using a calculator, so it is still important to know how to solve them manually.

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