10:08 PM Sat 1 Feb
Chapter 2 Electric Fie…
Assignment 1
4)

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Feb 01,2025

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10:08 PM Sat 1 Feb Chapter 2 Electric Fie... Assignment 1 4) $$ \int \frac{\cos (\ln x)}{x} d x= $$

UpStudy AI · Accepted Answer

To evaluate the integral:

$$
\int \frac{\cos (\ln x)}{x} \, dx
$$

**Step 1: Substitution**

Let’s make the substitution:

$$
t = \ln x \quad \Rightarrow \quad dt = \frac{1}{x} \, dx \quad \Rightarrow \quad dx = x \, dt
$$

Since $$ x = e^t $$, substituting into the integral gives:

$$
\int \frac{\cos(t)}{e^t} \cdot e^t \, dt = \int \cos(t) \, dt
$$

**Step 2: Integrate**

The integral of $$ \cos(t) $$ with respect to $$ t $$ is:

$$
\int \cos(t) \, dt = \sin(t) + C
$$

**Step 3: Substitute Back**

Replace $$ t $$ with $$ \ln x $$:

$$
\sin(t) + C = \sin(\ln x) + C
$$

**Final Answer:**

$$
\int \frac{\cos (\ln x)}{x} \, dx = \sin(\ln x) + C
$$

where $$ C $$ is the constant of integration.
The integral of $$ \frac{\cos (\ln x)}{x} $$ with respect to $$ x $$ is $$ \sin(\ln x) + C $$, where $$ C $$ is the constant of integration.

10:08 PM Sat 1 Feb
Chapter 2 Electric Fie…
Assignment 1
4)

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10:08 PM Sat 1 Feb Chapter 2 Electric Fie… Assignment 1 4)