integrate by parts
integrate (4+x^2)^1/2
To integrate the function (4 + x^2)^(1/2) using integration by parts, we need to follow these steps:
Step 1: Identify u and dv
Let's consider (4 + x^2)^(1/2) as f(x) for simplicity.
In integration by parts, we split the function into two parts: u and dv.
We choose u to be the part of the function that becomes simpler when differentiated, and dv to be the remaining part.
In this case, we can choose u = (4 + x^2)^(1/2) and dv = dx.
Step 2: Calculate du and v
Next, we need to calculate the differentials du and v.
To find du, we differentiate u with respect to x, and to find v, we integrate dv.
Differentiating u = (4 + x^2)^(1/2) with respect to x, we get:
du/dx = (1/2)(4 + x^2)^(-1/2)(2x)
du = (2x) / (2√(4 + x^2))
du = x / √(4 + x^2)
Integrating dv = dx, we get:
v = x
Step 3: Apply the integration by parts formula
The integration by parts formula is given as:
∫u * dv = uv - ∫v * du
Using the formula, we can rewrite the original integral as:
∫(4 + x^2)^(1/2) dx = u * v - ∫v * du
Substituting the values we calculated:
∫(4 + x^2)^(1/2) dx = (4 + x^2)^(1/2) * x - ∫x/√(4 + x^2) * dx
Step 4: Simplify the integral
Now, we have simplified the original integral into a new integral, which is still solvable:
∫x/√(4 + x^2) * dx
This new integral can be solved using a trigonometric substitution or a simpler substitution such as u = 4 + x^2.
By continuing these steps, you can solve the integral.
To integrate by parts, we will use the formula:
∫[u(x) * v'(x)] dx = u(x) * v(x) - ∫[v(x) * u'(x)] dx
Let's take u(x) = (4+x^2)^1/2 and v'(x) = 1. Then, we need to find v(x) by integrating v'(x):
v'(x) = 1
∫[v'(x)] dx = ∫[1] dx
v(x) = x + C where C is the constant of integration.
Now let's find u'(x):
u(x) = (4+x^2)^1/2
u'(x) = 1/2 * (4+x^2)^(-1/2) * 2x
u'(x) = x / (4+x^2)^1/2
Now we can use the integration by parts formula:
∫[(4+x^2)^1/2] dx = u(x) * v(x) - ∫[v(x) * u'(x)] dx
∫[(4+x^2)^1/2] dx = (4+x^2)^1/2 * (x + C) - ∫[(x + C) * (x / (4+x^2)^1/2)] dx
Now let's simplify the equation:
∫[(4+x^2)^1/2] dx = (4+x^2)^1/2 * (x + C) - ∫[(x^2 + Cx) / (4+x^2)^1/2] dx
To integrate ∫[(x^2 + Cx) / (4+x^2)^1/2] dx, we can use the substitution method. Let:
u = x^2 + 4
du = 2x dx
dx = du / (2x)
Now let's substitute:
∫[(x^2 + Cx) / (4+x^2)^1/2] dx = ∫[(u-4 + Cu) / u^1/2] * (du/ (2x))
Simplifying further, we get:
∫[(x^2 + Cx) / (4+x^2)^1/2] dx = ∫[(u-4 + Cu) / (u^1/2)] * (du/ (2x))
Now, we integrate:
∫[(u-4 + Cu) / (u^1/2)] * (du/ (2x))
= (1/2) * ∫[((u-4) / (u^1/2)) + (Cu / (u^1/2))] * (du/ (x))
= (1/2) * ∫[(u^1/2 - 4u^(-1/2)) + C(u^1/2 / (u^1/2))] * (du/ (x))
= (1/2) * ∫[(u^1/2 - 4u^(-1/2)) + C] * (du/ (x))
= (1/2) * [(2/3) u^(3/2) - 8u^(1/2) + C'x] + C''
Finally, substituting back u = x^2+4 and simplifying, we get:
∫[(4+x^2)^1/2] dx = (1/3) (4+x^2)^(3/2) - 4(4+x^2)^(1/2) + C'x + C''
Therefore, the integral of (4+x^2)^1/2 is given by:
(1/3) (4+x^2)^(3/2) - 4(4+x^2)^(1/2) + C'x + C''
where C' and C'' are constants of integration.