Question
Evaluate the integral
ln(2)
Alternative Form
≈0.693147
Evaluate
∫e2e3x(ln(x)−1)1dx
Multiply the terms
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
x(ln(x)−1)
Use the the distributive property to expand the expression
xln(x)+x(−1)
Multiplying or dividing an odd number of negative terms equals a negative
xln(x)−x
∫e2e3xln(x)−x1dx
Evaluate the integral
∫xln(x)−x1dx
Rewrite the expression
∫x(ln(x)−1)1dx
Use the substitution dx=xdt to transform the integral
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
t=ln(x)
Calculate the derivative
dt=x1dx
Evaluate
dx=xdt
∫x(ln(x)−1)1×xdt
Cancel out the common factor x
∫ln(x)−11dt
Use the substitution t=ln(x) to transform the integral
∫t−11dt
Use ∫ax+b1dx=a1ln∣ax+b∣ to evaluate the integral
ln(∣t−1∣)
Substitute back
ln(∣ln(x)−1∣)
Return the limits
(ln(∣ln(x)−1∣))e2e3
Solution
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Substitute the values into formula
ln(ln(e3)−1)−ln(ln(e2)−1)
Use lnen=n to simplify the expression
ln(∣3−1∣)−ln(ln(e2)−1)
Use lnen=n to simplify the expression
ln(∣3−1∣)−ln(∣2−1∣)
Subtract the numbers
ln(∣2∣)−ln(∣2−1∣)
Subtract the numbers
ln(∣2∣)−ln(∣1∣)
When the expression in absolute value bars is not negative, remove the bars
ln(2)−ln(∣1∣)
When the expression in absolute value bars is not negative, remove the bars
ln(2)−ln(1)
Removing 0 doesn't change the value,so remove it from the expression
ln(2)
ln(2)
Alternative Form
≈0.693147
Show Solution
![solution-arrow](data:image/svg+xml;base64,PHN2ZyB3aWR0aD0iMTciIGhlaWdodD0iMTYiIHZpZXdCb3g9IjAgMCAxNyAxNiIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPHBhdGggZD0iTTguNSAyLjY2Nzk3TDguNSAxMi42NjgiIHN0cm9rZT0id2hpdGUiIHN0cm9rZS13aWR0aD0iMS41IiBzdHJva2UtbGluZWNhcD0icm91bmQiIHN0cm9rZS1saW5lam9pbj0icm91bmQiLz4KPHBhdGggZD0iTTEyLjUgOC42Njc5N0w4LjUgMTIuNjY4TDQuNSA4LjY2Nzk3IiBzdHJva2U9IndoaXRlIiBzdHJva2Utd2lkdGg9IjEuNSIgc3Ryb2tlLWxpbmVjYXA9InJvdW5kIiBzdHJva2UtbGluZWpvaW49InJvdW5kIi8+Cjwvc3ZnPgo=)