i need to find the definite integral from 0 to 2|b| of ....
x divided by sqrt(x^2 + b^2) dx
im having a lot of trouble, thanks in advance
Let u = x^2 + b^2
du = 2x dx
The integrand becomes
(1/2) u^(-1/2) du
and the integral of this is u^(1/2)
= sqrt(x^2 + b^2)
Evaluate this between your limits.
First set u = x^2 + b^2
du = 2(x+b)or 2x + 2b
It would be very tedious to try to solve this on line with no signs for integral, radical, etc. But above is the first step to solving this problem. You must first find du. As you can see, you'll have some messy exponents to work with, something that would be very difficult to do here.
Is b a constant? This entire thing is crazy. Where is the dx?
If b is a constant,
du=2x dx
To solve the definite integral from 0 to 2|b| of (x divided by sqrt(x^2 + b^2)) dx, we can use the substitution method.
1. Let's start by substituting u = x^2 + b^2. This means du = 2x dx.
Rearranging this equation, we have dx = du / (2x).
2. Now let's substitute the expression for dx in terms of du and x into the integral:
∫ (x / sqrt(x^2 + b^2)) dx = ∫ ((1 / (2x)) * u^(-1/2)) du.
3. Simplifying the integrand further, we have:
(1 / 2) * ∫ (u^(-1/2) / x) du.
4. We can now evaluate the integral of u^(-1/2) with respect to u:
(1 / 2) * ∫ u^(-1/2) du = (1 / 2) * (2u^(1/2)) + C,
where C is the constant of integration.
5. Substituting back u = x^2 + b^2, we get:
(1 / 2) * (2u^(1/2)) + C = u^(1/2) + C = sqrt(x^2 + b^2) + C.
6. To find the definite integral between 0 and 2|b|, we evaluate the expression at the upper limit (2|b|) and subtract it from the expression evaluated at the lower limit (0):
[sqrt((2|b|)^2 + b^2) + C] - [sqrt((0)^2 + b^2) + C].
Note: The constant of integration cancels out because we are calculating the difference between two expressions.
So, the definite integral from 0 to 2|b| of (x / sqrt(x^2 + b^2)) dx is equal to:
sqrt((2|b|)^2 + b^2) - sqrt(b^2)