How do I solve this question using techniques from calculus?

Question

Integrate sin(x)sin(3x)/x^2 from -infinity to infinity

Answer 1

There are different ways to evaluate the integral

To evaluate the integral we can use the product-to-sum identities to simplify

sin⁡(a)sin⁡(b)=(1/2)[cos⁡(a−b)−cos⁡(a+b)]

Applying this to our integral gives:

sin⁡(x)sin⁡(3x)=(1/2)[cos⁡(2x)−cos⁡(4x)].

Now we can rewrite the integral:

I=∫_−∞^∞(1/2)(cos⁡(2x)/x²−cos⁡(4x)/x²)dx

This can be split into two separate integrals:

Since ∫_−∞^∞​cos(kx)/x^2​dx=π∣k∣.

Now you can separately evaluate each integral and get the final result.

Hope this helps.

If you still have difficulties, let me know.