Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. ∫0eln⁡x2dx

Question

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.  ∫0eln⁡x2dx

Provere · Accepted Answer

Step 1  Given definite integral is ∫0eln⁡x2dx.  Step 2  Evaluate the value of the integral as follows.  ∫0eln⁡x2dx=[xln⁡(x)2−∫2dx]0e  =[2xln⁡(x)−2x]0e  =2eln⁡(e)−2e−2(0)ln⁡(0)−2(0)  =2e-2e-0  =0  Thus, the given definite integral converges to 0.

Uersfeldte · Accepted Answer

Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:  ∫abf(x)dx=F(x)∣ab=F(b)−F(a)  Step 2: In this case, f(x)=ln⁡x2. Find its integral.  xln⁡x2−2((ln⁡x)x−x)∣0e  Step 3: Since F(x)∣ab=F(b)−F(a), expand the above into F(e)−F(0):  (eln⁡e2−2((ln⁡e)e−e))−(0ln⁡02−2(ln⁡0×0−0))  Step 4: Simplify.  =e

Determine whether the improper integral diverges or converges. Evaluate the

Answered question

Answer & Explanation

New Questions in Integral Calculus