Evaluate the integral. ∫arctan(√x) dx
Any idea how to get started?
if z = x^.5
dz = .5 x^-.5 dx
or
dx = 2 x^.5 dz = 2 z dz
and you have
2∫ z arctan(z) dz
but
http://www.wolframalpha.com/input/?i=integrate+z+tan%5E-1+z+dz
so
(z^2+1) arc tan (z) - z + constant
now put z = x^.5 back in
To evaluate the integral ∫arctan(√x) dx, we can use integration by parts.
Integration by parts is a method that allows us to evaluate certain types of integrals by rewriting them in a different form. It is based on the product rule for derivatives, which states that the derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.
Using integration by parts, we can express the integral of arctan(√x) as the product of two functions u and v:
∫arctan(√x) dx = u*v - ∫v*du
To determine which function to assign as u and v, we use the acronym LIATE, which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. We choose u to the function that appears earlier in this list and assign v to the other function.
In this case, arctan(√x) is the inverse trigonometric function and √x is the algebraic function. So, let's assign u = arctan(√x) and dv = dx.
To find du, the derivative of u, we can use the chain rule:
du = d(arctan(√x)) = (1/1+(√x)^2) * 1/(2√x) * dx
To find v, we can integrate dv = dx:
v = ∫dx = x
Now we have all the necessary components to evaluate the integral using the integration by parts formula:
∫arctan(√x) dx = u*v - ∫v*du
= arctan(√x) * x - ∫x * (1/1+(√x)^2) * 1/(2√x) * dx
Simplifying the expression:
= x * arctan(√x) - (1/2) * ∫dx / (1+x)
Now we can integrate the remaining term on the right-hand side:
= x * arctan(√x) - (1/2) * ln|1+x| + C
So, the integral of arctan(√x) dx is equal to x * arctan(√x) - (1/2) * ln|1+x| + C, where C is the constant of integration.