Evaluate the Integral

Problem

(∫_^)(cos(ln(x))/x*d(x))

Solution

1. Identify the substitution by noticing that the derivative of ln(x) is 1/x which is present in the integrand.

2. Substitute u=ln(x) which implies that d(u)=1/x*d(x)

3. Rewrite the integral in terms of u by replacing ln(x) with u and 1/x*d(x) with d(u)

(∫_^)(cos(u)*d(u))

4. Integrate the function with respect to u using the basic trigonometric integral rule (∫_^)(cos(u)*d(u))=sin(u)+C

sin(u)+C

5. Back-substitute the original expression for u to return to the variable x

sin(ln(x))+C

Final Answer

(∫_^)(cos(ln(x))/x*d(x))=sin(ln(x))+C

Want more problems? Check here!