The antiderivative of ln(2x+1) using integration by parts.
Take:
u = ln(2x+1)
du = 2/(2x+1) dx
dv = dx
v = x
|ln(2x+1) dx =
= xln(2x+1) - |2x/(2x+1) dx
= xln(2x+1) - |{1 - 1/(2x+1)} dx
= xln(2x+1) - |dx + |1/(2x+1) dx
= xln(2x+1) - x + (1/2)ln(2x+1) + const
To find the antiderivative of ln(2x+1) using integration by parts, we can follow these steps:
Step 1: Choose u and dv
Let u = ln(2x+1) and dv = dx.
Step 2: Find du and v
Differentiate u to find du:
du = (1 / (2x+1)) * 2 dx
du = dx / (2x+1)
Integrate dv to find v:
v = ∫ dx
v = x
Step 3: Apply the integration by parts formula
The integration by parts formula states: ∫ u dv = uv - ∫ v du.
Using this formula, we have:
∫ ln(2x+1) dx = (ln(2x+1)) * x - ∫ x * (dx / (2x+1))
Step 4: Simplify the remaining integral
Simplifying further, we have:
∫ x * (dx / (2x+1)) = ∫ (x / (2x+1)) dx
Step 5: Perform a u-substitution
Let u = 2x+1, then du = 2 dx.
Rewriting the integral in terms of u, we have:
∫ (x / (2x+1)) dx = ∫ (1/2) * (u-1) / u du
Simplifying this expression further, we have:
(1/2) * ∫ (u-1) / u du
Step 6: Evaluate the integral
∫ (u-1) / u du can be simplified into:
(1/2) * (∫ (u/u) du - ∫ (1/u) du)
(1/2) * (∫ 1 du - ∫ (1/u) du)
(1/2) * (u - ln|u|) + C
Step 7: Substitute u back in terms of x
Replacing u with 2x+1, we have:
(1/2) * ((2x+1) - ln|2x+1|) + C
Therefore, the antiderivative of ln(2x+1) using integration by parts is:
(1/2) * ((2x+1) - ln|2x+1|) + C, where C is the constant of integration.