Integration of sin (5x/2) / sin(x/2)

To integrate the given function, sin (5x/2) / sin(x/2), we can use the concept of trigonometric identities and substitution. Let's break down the integration process into steps:

1. Simplify the expression using trigonometric identities:
By using the identity sin(2θ) = 2sin(θ)cos(θ), we can rewrite the expression as:
sin (5x/2) / sin(x/2) = sin (5x/2) / (2sin(x/2)cos(x/2))

2. Apply the substitution method:
Let's substitute u = sin(x/2), then du = (1/2)cos(x/2)dx.
Rearranging the substitution equation, we have dx = (2/ cos(x/2))du.

3. Substitute the expressions involving x with u:
Now, we can rewrite the integral as:
∫ (sin (5x/2) / (2sin(x/2)cos(x/2))) dx = ∫ (sin (5x/2) / (2u(1 - u^2))) (2/ cos(x/2)) du

4. Simplify the expression:
By canceling out the 2's and rearranging the terms, we get:
∫ (sin (5x/2) / (u(1 - u^2))) (1/ cos(x/2)) du

5. Apply another trigonometric identity:
Using the identity sin(θ) = 2tan(θ/2) / (1 + tan²(θ/2)), we can rewrite the expression as:
∫ [(2tan((5x/2)/2) / (1 + tan²((5x/2)/2))] / [(u(1 - u^2)) / (1 + u^2)]

6. Simplify further:
By multiplying the fractions, we have:
∫ [2tan((5x/4)) / (1 + tan²((5x/4)))] / [(u(1 - u^2)) / (1 + u^2)] du

7. Integrate the expression:
Now, we can integrate the expression with respect to u, which will involve using the standard integral of tan(θ):
∫ [2tan((5x/4)) / (1 + tan²((5x/4)))] / [(u(1 - u^2)) / (1 + u^2)] du = ∫ [2tan((5x/4))] / [u(1 - u²)] du

The integral of tan(θ) is ln|sec(θ)| + C, where C is the constant of integration.

8. Finalize the integration:
Integrating the expression, we have:
∫ [2tan((5x/4))] / [u(1 - u²)] du = 2ln|sec((5x/4))| + C

Thus, the final result of integrating sin (5x/2) / sin(x/2) is 2ln|sec((5x/4))| + C, where C is the constant of integration.