# Is the Dirichlet integral a shortcut for solving this difficult integral?

• I
• Mr Davis 97
In summary, the conversation discusses using differentiation under the integral sign to solve for the integral ##\displaystyle \int_{- \infty}^{\infty} \frac{\cos x}{x^2+1} dx##. This involves manipulating the integral into a form where it can be linked to the Dirichlet integral. The question arises about how to handle the rightmost integral, which involves a variable ##t## in the argument of ##\sin##. The solution involves changing variables and recognizing the function as an even function, which allows for a link to the Dirichlet integral. The conversation also briefly mentions using the calculus of residues to solve for the original integral.
Mr Davis 97
I have the integral ##\displaystyle \int_{- \infty}^{\infty} \frac{\cos x}{x^2+1} dx##. We are going to use differentiation under the integral sign, so we let ##\displaystyle I(t) = \int_{- \infty}^{\infty} \frac{\cos tx}{x^2+1} dx##, and then, after manipulation, ##\displaystyle I'(t) = \int_{- \infty}^{\infty} \frac{\sin tx}{x(x^2+1)} dx - \int_{- \infty}^{\infty} \frac{\sin tx}{x} dx##. My question lies in the rightmost integral. In a solution I've seen, the rightmost integral is linked to the Dirichlet integral: https://en.wikipedia.org/wiki/Dirichlet_integral. And so ##\pi## is simply substituted for this expression. What I don't understand is how can it be linked to this known integral when there is that ##t## in the argument of ##\sin##?

Can you not change variables? Let ##u=tx##, etc. etc.

kuruman said:
Can you not change variables? Let ##u=tx##, etc. etc.
Okay, I see how that could work... But I don't know the sign of t, right? As such I can't tell whether the upper bound on the integral goes to ##+ \infty## or ##- \infty##

The final answer should not depend on the sign of ##t## because ##I(t)=I(-t)##. You have an even function in ##x## and ##t##. That helps you to link to the Dirchlet integral by putting a ##2## up front and taking the limits from zero to infinity.

Is there a reason for why you don't deal with the original integral via calculus of residues?

## 1. How can I determine which method to use when solving a difficult integral?

There are several methods that can be used to solve difficult integrals, including substitution, integration by parts, and trigonometric substitution. The best way to determine which method to use is to look for patterns and try different methods until one works. It may also be helpful to consult a calculus textbook or seek guidance from a math tutor.

## 2. Is there a specific order in which I should try different methods when solving a difficult integral?

There is no set order in which to try different methods, as it ultimately depends on the specific integral. However, it is generally recommended to start with substitution and then move on to integration by parts or trigonometric substitution if necessary.

## 3. How can I check my answer when solving a difficult integral?

One way to check your answer is to use an online integral calculator, which can verify the result and provide a step-by-step solution. You can also check your answer by differentiating it and comparing it to the original integrand.

## 4. Are there any tips for making the process of solving a difficult integral easier?

One helpful tip is to carefully look for any algebraic manipulations or trigonometric identities that can simplify the integral before attempting to solve it. It can also be useful to practice and familiarize yourself with different integration techniques.

## 5. What should I do if I am unable to solve a difficult integral?

If you are unable to solve a difficult integral, do not get discouraged. You can always seek help from a math tutor or consult with other mathematicians. In some cases, the integral may not have a closed form solution, and numerical methods may need to be used instead.

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