Evaluate:
integral x^3/[sqrt*(x^2+100)]dx
can someone tell me what they get as a final answer..I've been going at this for hours and don't get the answer in the back of the textbook
How about
(1/3)(x^2 - 200) / √(x^2 + 100)
I checked it by differentiating it.
there should be no division, I copied the verification, silly me
go with (1/3)(x^2 - 200)√(x^2 + 100)
which btw is what "helper" also had in the other post.
To solve this integral, you can start by applying a substitution u = x^2 + 100. Let's walk through the steps together.
1. Start by differentiating u = x^2 + 100 with respect to x:
du/dx = 2x
2. Rearrange the equation from step 1 to solve for dx in terms of du:
dx = du / (2x)
3. Substitute x^3 as (u - 100) into the integral and replace dx with du / (2x):
∫ (u - 100) / [sqrt(u)] * (du / (2x))
4. Simplify the expression using the properties of exponents:
∫ [(u - 100) / (sqrt(u))] * (1 / (2x)) * du
5. Split the fraction into two parts:
∫ (u / (2x * sqrt(u))) - (100 / (2x * sqrt(u))) du
6. Using properties of integrals, split the integral into two parts:
∫ (u / (2x * sqrt(u))) du - ∫ (100 / (2x * sqrt(u))) du
7. Now, integrate each part separately:
a. For the first integral, we can simplify (u / (2x * sqrt(u))) to (1 / 2x) * sqrt(u):
∫ (1 / 2x) * sqrt(u) du
Apply the power rule for integration: integrate x^n with respect to x to get (x^(n+1))/(n+1):
(1 / 2x) * 2/3 * u^(3/2) + C [where C is the constant of integration]
Simplify the expression:
(1 / 3x) * u^(3/2) + C
b. For the second integral, we have a constant factor, so it can be brought out of the integral:
- (100 / (2x)) * ∫ (1 / sqrt(u)) du
Apply the rule of integrating 1/x, which states that the integral of (1/x) dx is ln|x| + C:
- (100 / (2x)) * 2sqrt(u) + C
Simplify the expression:
- (100 / x) * sqrt(u) + C
8. Putting the results of the two integrals together, we have:
∫ (x^3) / sqrt(x^2 + 100) dx = (1 / 3x) * u^(3/2) - (100 / x) * sqrt(u) + C
Remember to substitute back u = x^2 + 100 into the final result.
Note: It's essential to double-check every step for possible mistakes.