evaluate the integral or state that it diverges. Check if I did it correctly.
integral 0,1 dr/r^.999
lim
b->0+ integral b, 1 1000r^.001 =-1000
To evaluate the integral, we can use the properties of logarithms and the power rule for integrals. Let's break down the integral step by step:
∫(0 to 1) dr/r^0.999
First, let's simplify the exponent:
∫(0 to 1) dr/r^(0.999)
Now, let's apply the power rule for integrals:
∫(0 to 1) r^(-0.999) dr
To integrate, we can add 1 to the exponent and divide by the new exponent:
[ r^(-0.999 + 1) ] / (-0.999 + 1) from 0 to 1
Simplifying further:
[ r^0.001 ] / (0.001) from 0 to 1
Now, let's plug in the upper limit (1) into the expression:
[ 1^0.001 ] / 0.001
Since any non-zero number raised to the power of 0.001 is still 1, we have:
[ 1 ] / 0.001
Which simplifies to:
1 / 0.001 = 1000
Now, let's plug in the lower limit (0) into the expression:
[ 0^0.001 ] / 0.001
Any non-zero number raised to the power of 0 is 1:
[ 0 ] / 0.001
Since division by zero is undefined, the lower limit gives us an indeterminate form.
Finally, let's evaluate the overall integral:
∫(0 to 1) dr/r^0.999 = 1000 - undefined
Therefore, the integral does not converge or have a definite value because the lower limit gives us an indeterminate form.