Integration of sin x / sin3x

To integrate the function sin x / sin 3x, we can use the technique of trigonometric substitutions. Let's break down the steps involved in solving this integral.

1. Simplify the Function:
Start by simplifying the given function, sin x / sin 3x. We can use the trigonometric identity sin 3x = 3sin x - 4sin^3 x to rewrite the function as sin x / (3sin x - 4sin^3 x).

2. Apply Trigonometric Substitution:
To simplify the integral further, we can make a substitution. Let's substitute u = sin x. This will transform the integral into a more manageable form.

3. Express dx in terms of du:
To express dx in terms of du, we differentiate both sides of the substitution equation with respect to x. Since u = sin x, du/dx = cos x. Rearranging this equation, we get dx = du / cos x.

4. Rewrite the Integral:
Using the substitution and the expression for dx, we can rewrite the integral as ∫ (1 / (3u - 4u^3)) * (du / cos x).

5. Simplify the Integral:
Next, we need to simplify the integral further. Recall that u = sin x. Therefore, we can rewrite the integral as ∫ (1 / (3u - 4u^3)) * (du / cos x) = ∫ (1 / (3u - 4u^3)) * (1 / cos x) du.

6. Apply Trigonometric Identity:
Now, we can apply the trigonometric identity cos x = √(1 - sin^2 x) to simplify the integral. Substituting this identity, we have ∫ (1 / (3u - 4u^3)) * (1 / √(1 - sin^2 x)) du.

7. Simplify the Integral Further:
To simplify the integral even more, we can use the trigonometric identity sin^2 x + cos^2 x = 1. Rearranging this identity, we get sin^2 x = 1 - cos^2 x. Substituting this into the integral, we have ∫ (1 / (3u - 4u^3)) * (1 / √(cos^2 x)) du.

8. Express the Integral in terms of u:
After simplifying, the integral becomes ∫ (1 / (3u - 4u^3)) * (1 / |cos x|) du.

9. Integrate the Function:
Finally, we can integrate the function ∫ (1 / (3u - 4u^3)) * (1 / |cos x|) du. This integration process may involve additional techniques such as partial fractions or the method of substitution, depending on the specific form of the function.

By following these steps, we can integrate the function sin x / sin 3x and find its antiderivative. Remember to evaluate the constant of integration after integrating the function.