Evaluate the definite integral.∫from 0 to 2 x/√(1+2x²)dx
To evaluate the definite integral ∫(0 to 2) (x/√(1+2x²)) dx, we can use a technique called substitution. Substitute u = 1 + 2x² for simplicity.
First, find du/dx and solve for dx:
du/dx = 4x
dx = du / 4x
Next, substitute the value of dx into the integral:
∫(0 to 2) [x/√(1+2x²)] dx = ∫(0 to 2) [x/√u] (du/4x)
Now, simplify the integral:
∫(0 to 2) (1/4√u) du
Since the left side of the integral is independent of u, we can simplify the integral further:
(1/4) ∫(0 to 2) (1/√u) du
Now, apply the power rule of integrals:
(1/4) [2√u] (from 0 to 2)
Evaluate at the upper and lower limits:
(1/4) [(2√2) - (2√0)]
Simplify the result:
(1/4) [2√2 - 0] = (1/4) * 2√2 = √2/2
Therefore, the value of the definite integral ∫(0 to 2) (x/√(1+2x²)) dx is √2/2.