If `int _(0)^(oo) (cos x)/(x ) dx =pi/2 , then int _(0)^(oo) (cos ^(3) x )/(x) dx` is equals to:

Options

A. `pi/2`
B. `pi/4`
C. `pi`
D. `(3pi)/(2)`

Answer

To solve the integral \( \int_{0}^{\infty} \frac{\cos^3 x}{x} \, dx \) given that \( \int_{0}^{\infty} \frac{\cos x}{x} \, dx = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Use the identity for \( \cos^3 x \)

We know the identity:
\[ \cos 3x = 4 \cos^3 x - 3 \cos x \]
From this, we can express \( \cos^3 x \) in terms of \( \cos 3x \) and \( \cos x \):
\[ \cos^3 x = \frac{\cos 3x + 3 \cos x}{4} \]

### Step 2: Substitute into the integral

Substituting this expression into the integral, we have:
\[ \int_{0}^{\infty} \frac{\cos^3 x}{x} \, dx = \int_{0}^{\infty} \frac{1}{x} \left( \frac{\cos 3x + 3 \cos x}{4} \right) \, dx \]
This simplifies to:
\[ \int_{0}^{\infty} \frac{\cos^3 x}{x} \, dx = \frac{1}{4} \int_{0}^{\infty} \frac{\cos 3x}{x} \, dx + \frac{3}{4} \int_{0}^{\infty} \frac{\cos x}{x} \, dx \]

### Step 3: Evaluate the integrals

We know from the problem statement that:
\[ \int_{0}^{\infty} \frac{\cos x}{x} \, dx = \frac{\pi}{2} \]
Now, we need to evaluate \( \int_{0}^{\infty} \frac{\cos 3x}{x} \, dx \). By using the substitution \( t = 3x \), we have \( dt = 3dx \) or \( dx = \frac{dt}{3} \). The limits remain the same (0 to \( \infty \)):
\[ \int_{0}^{\infty} \frac{\cos 3x}{x} \, dx = \int_{0}^{\infty} \frac{\cos t}{t} \, dt = \frac{\pi}{2} \]

### Step 4: Substitute back into the integral

Now substituting back:
\[ \int_{0}^{\infty} \frac{\cos^3 x}{x} \, dx = \frac{1}{4} \cdot \frac{\pi}{2} + \frac{3}{4} \cdot \frac{\pi}{2} \]
This simplifies to:
\[ \int_{0}^{\infty} \frac{\cos^3 x}{x} \, dx = \frac{\pi}{8} + \frac{3\pi}{8} = \frac{4\pi}{8} = \frac{\pi}{2} \]

### Final Answer

Thus, the value of \( \int_{0}^{\infty} \frac{\cos^3 x}{x} \, dx \) is:
\[ \boxed{\frac{\pi}{2}} \]