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

Answer 1

Use the u substitution.

u = #sqrt(x)#

du = #1/(2sqrt(x))# dx

2du = #1/sqrt(x)# dx

Write the new formula after the u substitution.

2 #int# #tan^-1(u)# du

Use table 89 to find the integral of 2#tan^-1(u)#.

2 #int# #tan^-1(u)# du
= 2[u #tan^-1(u)# - #1/2# ln(1 + #u^2#)] + C

Replace the u variable back in the terms of x.

= 2[#sqrt(x)# #tan^-1(sqrt(x))# - #1/2# ln(1 + #sqrt(x)^2#)] + C

Simplify the answer.

= 2[#sqrt(x)# #tan^-1(sqrt(x))# - #1/2# ln(1 + #x#)] + C

= 2#sqrt(x)# #tan^-1(sqrt(x))# - ln(1 + #x#) + C

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Answer 2

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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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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