To evaluate the given indefinite integral, \(\int \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx\), we will use a common technique called the substitution method. This method helps simplify complex integrals into a more manageable form.

Evaluating the Integral: \( \int \sin(\sqrt{x})/\sqrt{x} \, dx \)

The integral we need to solve is:

\[ \int \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx \]

Substitution Method for Integration

The key to solving this integral is to identify a suitable substitution that simplifies the expression. We observe that the derivative of \(\sqrt{x}\) is related to \(\frac{1}{\sqrt{x}}\), which appears in the denominator.

• Let \(u\) be equal to \(\sqrt{x}\).
• So, \(u = \sqrt{x}\).
• Next, we need to find the differential \(du\) in terms of \(dx\). We differentiate \(u\) with respect to \(x\):
• \( \frac{du}{dx} = \frac{d}{dx}(\sqrt{x}) \)
• We know that \(\sqrt{x}\) can be written as \(x^{1/2}\). Therefore, its derivative using the power rule is:
• \( \frac{du}{dx} = \frac{1}{2} x^{(1/2 - 1)} = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \)
• From this, we can express the term \(\frac{1}{\sqrt{x}} \, dx\) in terms of \(du\):
• \( du = \frac{1}{2\sqrt{x}} \, dx \)
• Multiplying both sides by 2, we get:
• \( 2 \, du = \frac{1}{\sqrt{x}} \, dx \)

Transforming the Integral Expression

Now, we substitute \(u = \sqrt{x}\) and \(2 \, du = \frac{1}{\sqrt{x}} \, dx\) into the original integral. The integral transforms as follows:

\[ \int \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx = \int \sin(u) \cdot \left(\frac{1}{\sqrt{x}} \, dx\right) \]

Substitute the terms into the integral:

\[ = \int \sin(u) \cdot (2 \, du) \]

We can pull the constant factor (2) out of the integral sign:

\[ = 2 \int \sin(u) \, du \]

Integrating the Transformed Expression

Now, we need to integrate \(\sin(u)\) with respect to \(u\). The standard integral of \(\sin(x)\) is known to be \(-\cos(x)\).

• The integral of \(\sin(u)\) is \(-\cos(u)\).
• So, the transformed expression becomes:
• \( 2 \int \sin(u) \, du = 2(-\cos(u)) + C \)
• \( = -2\cos(u) + C \)

Here, \(C\) represents the constant of integration, which is always added for indefinite integrals.

Back-Substitution to Original Variable

The final step is to substitute back the original variable \(x\) into our result. We initially defined \(u = \sqrt{x}\).

• Replace \(u\) with \(\sqrt{x}\) in the obtained result:
• \( -2\cos(u) + C = -2\cos(\sqrt{x}) + C \)

Thus, the evaluation of the integral \(\int \sin(\sqrt{x})/\sqrt{x} \, dx\) is \(-2\cos(\sqrt{x}) + C\).

Comparing with Options

Let's compare our derived solution with the provided multiple-choice options:

• Option 1: cos \(\sqrt x\) + c
• Option 2: -2 cosec \(\sqrt x\) + c
• Option 3: -2 cos \(\sqrt x\) + c
• Option 4: sec \(\sqrt x\) + c

Our calculated result, \(-2\cos(\sqrt{x}) + C\), precisely matches Option 3.