# How do you evaluate #int# #arctan(sqrt(x))/sqrt(x)# dx?

##
Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)

#int# #arctan(sqrt(x))/sqrt(x)# dx

Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)

Use the u substitution.

u =

du =

2du =

Write the new formula after the u substitution.

2

Use table 89 to find the integral of 2

2

= 2[u

Replace the u variable back in the terms of x.

= 2[

Simplify the answer.

= 2[

= 2

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To evaluate the integral of arctan(sqrt(x))/sqrt(x) dx, we can use a substitution method. Let's make the substitution u = sqrt(x), then du = (1/2)sqrt(x) dx. Rearranging this equation gives us dx = 2u du. Substituting these into the integral:

∫arctan(u) du

Now, let's integrate by parts:

Integration: ∫arctan(u) du = u * arctan(u) - ∫(1 + u^2) / (1 + u^2) du = u * arctan(u) - ∫du

Integrating 1 with respect to u gives us u, so the integral becomes:

u * arctan(u) - u + C

Substitute back u = sqrt(x):

sqrt(x) * arctan(sqrt(x)) - sqrt(x) + C

So, the integral evaluates to sqrt(x) * arctan(sqrt(x)) - sqrt(x) + C, where C is the constant of integration.

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