integral of xarctanxdx using integration by parts.
∫x arctan x dx
u = arctan x
du = 1/(1+x^2)
dv = x dx
v = 1/2 x^2
∫ x arctan x dx = 1/2 x^2 arctan x - ∫ 1/2 x^2 * 1/(1+x^2) dx
= 1/2 (x^2 arctan(x) - ∫x^2/(1+x^2) dx)
now, x^2/(1+x^2) = 1 - 1/(1+x^2)
∫x^2/(1+x^2) dx = ∫ 1 - 1/(1+x^2) dx
= x - arctan(x)
so we wind up with
1/2 (x^2 arctan(x) - (x - arctan(x)))
= 1/2 ((1+x^2)arctan(x) - x) + C
∫xarctan(x) dx
Integration by parts:
∫vdu = vu - ∫udv
Where:
v = arctan(x) du = xdx
dv = 1/(x^2 + 1)dx u = x^2/2
∫xarctan(x)dx = (x^2/2)arctan(x) - ∫(x^2)/2(x^2 + 1)dx
Factor out constants:
(1/2)x^2arctan(x) - (1/2)∫x^2/(x^2+1)dx
Long division in the integrand:
(1/2)x^2arctan(x) - (1/2)∫(1-(1/(x^2+1))dx
Separate integrand:
(1/2)x^2arctan(x) - (1/2)∫1dx -(1/2)∫1/(x^2+1)dx
Integrate:
(1/2)x^2arctan(x) - (x/2) - (1/2)arctan(x) + C
Factor:
(1/2)(x^2arctan(x) - x - arctan(x)) + C
Complete:
∫xarctan(x)dx = (1/2)(x^2arctan(x)-x-arctan(x)) + C
thank you so much! I was getting stuck at the long division part.
To find the integral of x * arctan(x) dx using integration by parts, we'll need to follow the integration by parts formula:
∫ u dv = uv - ∫ v du,
where u and v represent two functions of x. In our case, we'll let u = arctan(x) and dv = x dx.
Step 1: Differentiate u
To find du (the derivative of u), we'll use the chain rule:
du = (1 / (1 + x^2)) * dx.
Step 2: Integrate dv
To find v (the integral of dv), we'll integrate x dx:
∫ x dx = (1/2) x^2 + C.
Now that we have u, du, v, and dv, we can substitute them into the integration by parts formula:
∫ x * arctan(x) dx = uv - ∫ v du,
∫ x * arctan(x) dx = arctan(x) * [(1/2) x^2 + C] - ∫ [(1/2) x^2 + C] * (1 / (1 + x^2)) dx.
Simplifying this expression, we get:
∫ x * arctan(x) dx = (1/2) x^2 * arctan(x) + C1 - (1/2) ∫ x^2 / (1 + x^2) dx - ∫ C / (1 + x^2) dx.
Now, let's solve each integral separately:
∫ x^2 / (1 + x^2) dx:
To solve this integral, we can use a substitution. Let u = 1 + x^2, then du = 2x dx. Rearranging the terms, dx = (1 / 2x) du. Substituting back into the integral:
∫ x^2 / (1 + x^2) dx = ∫ (x^2 * (1 / 2x)) du,
∫ x^2 / (1 + x^2) dx = (1/2) ∫ 1 du,
∫ x^2 / (1 + x^2) dx = (1/2) u + C2.
∫ C / (1 + x^2) dx:
This integral is straightforward since C is just a constant. It integrates to:
∫ C / (1 + x^2) dx = C ∫ 1 / (1 + x^2) dx,
∫ C / (1 + x^2) dx = C arctan(x) + C3.
Substituting all our results back into the previous expression, we get:
∫ x * arctan(x) dx = (1/2) x^2 * arctan(x) + C1 - (1/4) u - C2 arctan(x) - C3,
Simplifying further,
∫ x * arctan(x) dx = (1/2) x^2 * arctan(x) - (1/4) (1 + x^2) - C2 arctan(x) - C3,
which can be written as:
∫ x * arctan(x) dx = (1/2) x^2 * arctan(x) - (1/4) - (1/4) x^2 - C2 arctan(x) - C3.
So, the final answer is (1/2) x^2 * arctan(x) - (1/4) - (1/4) x^2 - C2 arctan(x) - C3, where C1, C2, and C3 are constants of integration.