Question

$I = \int \cos(\ln x) \, dx$. Then $I =$

• A. x/2 {cos (In x) + sin ( In x) } +c
• B. x2 {cos (In x) - sin ( In x ) }+c
• C. x2 sin (In x )+c
• D. x cos ( In x) +c ( c denotes constant of integration)

Answer 1

The Correct Option is A

We are tasked with evaluating the integral: \[ I = \int \cos(\ln x) \, dx \] Step 1: Use substitution Let’s begin by using the substitution method. Let: \[ u = \ln x \] Then: \[ du = \frac{1}{x} \, dx \] Thus, \( dx = x \, du \), and since \( x = e^u \), the integral becomes: \[ I = \int \cos(u) \cdot e^u \, du \] Step 2: Solve the new integral Now, we need to evaluate: \[ \int e^u \cos(u) \, du \] This is a standard integral that can be solved using integration by parts or directly using the known formula: \[ \int e^u \cos(u) \, du = \frac{e^u}{2} \left( \cos(u) + \sin(u) \right) \] Step 3: Substitute back in terms of \( x \) Now, substitute \( u = \ln x \) back into the result: \[ I = \frac{x}{2} \left( \cos(\ln x) + \sin(\ln x) \right) + C \] where \( C \) is the constant of integration. Conclusion Thus, the solution is: \[ I = \frac{x}{2} \left( \cos(\ln x) + \sin(\ln x) \right) + C \] The correct answer is: \[ \boxed{\frac{x}{2} \left( \cos(\ln x) + \sin(\ln x) \right) + C} \]