Question
Evaluate the integral
1
Evaluate
∫1ex×xln(x)xln(x)dx
Reduce the fraction
More Steps

Evaluate
x×xln(x)xln(x)
Multiply the terms
x2ln(x)xln(x)
Reduce the fraction
More Steps

Calculate
x2x
Use the product rule aman=an−m to simplify the expression
x2−11
Subtract the terms
x11
Simplify
x1
xln(x)ln(x)
Reduce the fraction
x1
∫1ex1dx
By definition,rewrite the improper integral using one-sided limit and a definite integral
a→1+lim(∫aex1dx)
Evaluate the integral
More Steps

Evaluate
∫aex1dx
Evaluate the integral
∫x1dx
Use the property of integral ∫x1dx=ln∣x∣
ln(∣x∣)
Return the limits
(ln(∣x∣))ae
Substitute the values into formula
ln(∣e∣)−ln(∣a∣)
When the expression in absolute value bars is not negative, remove the bars
ln(e)−ln(∣a∣)
A logarithm with the same base and argument equals 1
1−ln(∣a∣)
a→1+lim(1−ln(∣a∣))
Rewrite the expression
a→1+lim(1)+a→1+lim(−ln(∣a∣))
Calculate
1+a→1+lim(−ln(∣a∣))
Any expression multiplied by 0 equals 0
More Steps

Evaluate
a→1+lim(−ln(∣a∣))
Rewrite the expression
−ln(a→1+lim(∣a∣))
Calculate
More Steps

Calculate
ln(a→1+lim(∣a∣))
Calculate
ln(1)
Evaluate the logarithm
0
0
1+0
Solution
1
Show Solution
