Integration of √x sin(x) dx

To integrate the function √x sin(x) dx, we can use integration by parts. Integration by parts is a technique that allows us to rewrite the integral of a product of two functions as a new integral that is hopefully easier to evaluate. The formula for integration by parts is ∫ u dv = uv - ∫ v du, where u and v are functions of x.

Step 1: Identify u and dv
In order to use integration by parts, we need to choose u and dv. Let's choose:

u = √x
dv = sin(x) dx

Step 2: Find du and v
Now, we need to find du and v by differentiating u and integrating dv, respectively:

du = (1/2√x) dx
v = -cos(x)

Step 3: Apply the integration by parts formula
Using the integration by parts formula, we can rewrite the integral as:

∫ √x sin(x) dx = -√x cos(x) - ∫ (-cos(x))(1/2√x) dx

Simplifying this expression gives:

∫ √x sin(x) dx = -√x cos(x) + (1/2) ∫ (cos(x)/√x) dx

Step 4: Evaluate the remaining integral
The new integral, ∫ (cos(x)/√x) dx, can be evaluated using a substitution. Let's choose u = √x, then du = (1/2√x) dx. Substituting these values into the integral, we get:

(1/2) ∫ (cos(x)/√x) dx = (1/2) ∫ cos(x) du

Integrating cos(x) with respect to u gives:

(1/2) ∫ cos(x) du = (1/2) sin(x) + C

Step 5: Finalize the solution
Now, we can substitute this result back into our original expression:

∫ √x sin(x) dx = -√x cos(x) + (1/2) sin(x) + C

Therefore, the integral of √x sin(x) dx is equal to -√x cos(x) + (1/2) sin(x) + C, where C is the constant of integration.

This is the step-by-step process to integrate the function √x sin(x) dx using the method of integration by parts.