Evaluate each indefinite integral. Use the substitution method.
X square root (x+3) dx
To solve the indefinite integral ∫x√(x+3) dx using the substitution method, you can follow these steps:
1. Let u = x+3
By making this substitution, we can rewrite the integral in terms of the new variable u.
2. Find the derivative of u with respect to x.
du/dx = 1
3. Solve for dx in terms of du.
dx = du
4. Substitute u and dx in terms of u in the original integral.
∫x√(x+3) dx = ∫(u-3)√u du
5. Simplify the integral.
∫(u-3)√u du = ∫(u^(3/2) - 3u^(1/2)) du
6. Integrate each term separately.
∫u^(3/2) du = (2/5)u^(5/2) + C1
∫3u^(1/2) du = 2u^(3/2) + C2
7. Combine the results and include the constant of integration.
∫x√(x+3) dx = (2/5)(x+3)^(5/2) + 2(x+3)^(3/2) + C
Therefore, the final solution to the indefinite integral ∫x√(x+3) dx using the substitution method is (2/5)(x+3)^(5/2) + 2(x+3)^(3/2) + C, where C represents the constant of integration.