Find the integral of:
2xsqrt(x+1) dx
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let u = 2x
du = 2 dx
let dv = (x+1)^(1/2) dx
v = (2/3)(x+1)^(3/2)
∫2xsqrt(x+1) dx = uv - ∫v du
= (2x(2/3)(x+1)^(3/2) - ∫(4/3)(x+1)^(3/2) dx
= (4/3)x(x+1)^(3/2) - (8/15)(x+1)^(5/2)
= (1/15)(x+1)^(3/2) (20x - 8(x+1) )
= (4/15)(x+1)^(3/2) (5x - 2x - 2)
= (4/15)(x+1)^(3/2) (3x - 2)
To find the integral of 2x√(x+1) dx, we can use a technique called integration by substitution. Here's how you can do it:
Step 1: Identify a suitable substitution.
Let's substitute u = x+1. This choice will help us simplify the expression and make it easier to integrate.
Step 2: Find the differential of the substitution.
Taking the derivative of both sides with respect to x, we get du/dx = 1. Rearranging, we have du = dx.
Step 3: Rewrite the integral with the substitution.
Using the substitution u = x+1, we can rewrite the integral as:
∫ 2x√(x+1) dx = ∫ 2(u-1)√u du.
Step 4: Simplify and solve the integral with the new variable.
Expanding the expression inside the integral, we have:
∫ 2(u-1)√u du = ∫ (2u√u - 2√u) du.
Now we can integrate each term separately:
∫ 2u√u du = 2∫ u^(3/2) du, which can be solved using the power rule of integration.
∫ 2√u du = 2∫ u^(1/2) du, which can also be solved using the power rule of integration.
Step 5: Evaluate the integrals.
∫ u^(3/2) du = (2/5)u^(5/2) + C1, where C1 is the constant of integration.
∫ u^(1/2) du = (4/3)u^(3/2) + C2, where C2 is the constant of integration.
Step 6: Substitute the original variable back in.
Using the original substitution u = x+1, we can substitute back to get the final result:
∫ 2x√(x+1) dx = (2/5)(x+1)^(5/2) + C1 - (8/3)(x+1)^(3/2) + C2.
So, the integral of 2x√(x+1) dx is (2/5)(x+1)^(5/2) - (8/3)(x+1)^(3/2) + C, where C = C1 + C2 is the constant of integration.