If the substitution u=x^2 + 1 is made, then the integral from 0 to 2 of [(x^3)/(x^2 + 1)]dx = ?
Thank you for your help!
To solve this integral using the given substitution u = x^2 + 1, we need to express the integral in terms of u. Here's how to do it step by step:
1. Start with the given integral: ∫[(x^3)/(x^2 + 1)]dx, with the limits of integration from 0 to 2.
2. Substitute u = x^2 + 1. To do this, we need to find dx in terms of du. Take the derivative of both sides of the substitution equation:
du/dx = d/dx(x^2 + 1)
du/dx = 2x
dx = du/(2x)
3. Rewrite the integral with the new variable u and dx:
∫[(x^3)/(x^2 + 1)]dx = ∫[((u - 1)/(2x)) * x^3]dx
= ∫[(u - 1)/2 * x^2]dx
= ∫[(u - 1)/2 * (u - 1 - 1)]dx
= ∫[(u - 1)^2/2]dx
4. Now, we need to change the limits of integration. When x = 0, u = x^2 + 1 = 1. And when x = 2, u = x^2 + 1 = 2^2 + 1 = 5. So the new limits of integration are from 1 to 5.
5. The integral becomes:
∫[(u - 1)^2/2]dx = ∫[(u - 1)^2/2]du (since dx = du/(2x) = du/(2√u - 2))
= ∫[(u^2 - 2u + 1)/2]du
= (1/2)∫[u^2 - 2u + 1]du
= (1/2)[(u^3/3) - u^2/2 + u] + C
= (1/6)u^3 - (1/4)u^2 + (1/2)u + C
6. Finally, evaluate the integral at the new limits of integration:
∫[(x^3)/(x^2 + 1)]dx from 0 to 2
= [(1/6)u^3 - (1/4)u^2 + (1/2)u] from 1 to 5
= [(1/6)(5^3) - (1/4)(5^2) + (1/2)(5)] - [(1/6)(1^3) - (1/4)(1^2) + (1/2)(1)]
= [125/6 - 25/4 + 5/2] - [1/6 - 1/4 + 1/2]
= [(125 - 75 + 60)/12] - [(2 - 3 + 6)/12]
= (110/12) - (5/12)
= 105/12
So, the value of the integral from 0 to 2 of [(x^3)/(x^2 + 1)]dx is 105/12.