3.3 Q-19

Question Statement

Evaluate the following integral:

$$\int \left[ \cos \left( \sqrt{x} - \frac{x}{2} \right) \right] \times \left[ \frac{1}{\sqrt{x}} - 1 \right] , dx$$

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Background and Explanation

To solve this integral, we will use a substitution method to simplify the expression. The integrand consists of a cosine function multiplied by a rational expression. A useful substitution will make the cosine function easier to handle and transform the integral into a simpler form.

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Solution

Let’s solve the integral step by step:

1. Substitution: We begin by setting the expression inside the cosine function as a new variable:

$$\sqrt{x} - \frac{x}{2} = u$$

Now, differentiate both sides with respect to $x$:

$$\frac{d}{dx} \left( \sqrt{x} - \frac{x}{2} \right) = \frac{1}{2\sqrt{x}} - \frac{1}{2}$$

Therefore, the differential $dx$ becomes:

$$dx = 2 , du$$

2. Transform the Integral: Substituting $u = \sqrt{x} - \frac{x}{2}$ into the original integral, we also replace the differential $dx$ as $2 , du$. The term $\frac{1}{\sqrt{x}} - 1$ becomes part of $du$, so we have:

$$\int \left[ \cos \left( \sqrt{x} - \frac{x}{2} \right) \right] \times \left[ \frac{1}{\sqrt{x}} - 1 \right] , dx = \int \cos(u) \times 2 , du$$

3. Simplify the Integral: The integral becomes:

$$2 \int \cos(u) , du$$

4. Integrate: The integral of $\cos(u)$ is $\sin(u)$, so we get:

$$2 \sin(u) + C$$

5. Substitute Back: Finally, substitute back the expression for $u$:

$$u = \sqrt{x} - \frac{x}{2}$$

Therefore, the solution to the integral is:

$$2 \sin \left( \sqrt{x} - \frac{x}{2} \right) + C$$

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Key Formulas or Methods Used

• Substitution Method: The substitution $u = \sqrt{x} - \frac{x}{2}$ helped simplify the integral.
• Basic Integration: The integral of $\cos(u)$ is $\sin(u)$.

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Summary of Steps

1. Substitute $\sqrt{x} - \frac{x}{2} = u$ to simplify the expression.
2. Differentiate to find $dx = 2 , du$.
3. Substitute into the integral, resulting in $2 \int \cos(u) , du$.
4. Integrate to get $2 \sin(u) + C$.
5. Substitute $u = \sqrt{x} - \frac{x}{2}$ back into the solution:

$$2 \sin \left( \sqrt{x} - \frac{x}{2} \right) + C$$