the integral from 0 to lnx of lnx/(x^1/2)
thanks!
To evaluate the integral, we can use the technique of integration by substitution. Here's how to do it step by step:
Step 1: First, let's choose a substitution. Since we have an ln(x) term in the numerator and an x^(-1/2) term in the denominator, a suitable substitution would be u = ln(x). This implies that du/dx = 1/x, or dx = x du.
Step 2: Next, we need to change the limits of integration. The original limits are 0 and lnx. To find the new limits, we substitute the original limits into our chosen substitution. When x = 0, ln(x) is undefined, so the lower limit becomes u = ln(0), which is undefined. When x = lnx, we have u = ln(lnx).
Step 3: Now, let's rewrite the integral using our substitution:
∫[0, lnx] ln(x)/x^(1/2) dx = ∫[undefined, ln(lnx)] ln(e^u)/(e^u)^(1/2) x du = ∫[undefined, ln(lnx)] u/e^(u/2) du
Step 4: We can simplify further by separating the variables:
∫[undefined, ln(lnx)] u/e^(u/2) du = ∫[undefined, ln(lnx)] u e^(-u/2) du
Step 5: At this point, we can integrate the resulting function. Note that the integral of u e^(-u/2) du can be evaluated using integration by parts.
Step 6: After performing the integration, we obtain the final result in terms of the original variable x.
It's worth noting that the integral involves an undefined lower limit of integration, which makes it impossible to obtain a definite numerical value for the integral. However, by following the steps outlined above, you can reach the solution in terms of the given function.