Integrate x^2 / (x^2 + a^2)^3/2 dx

[Ray Beaver]

Homework Statement

My solution has two terms divided by a which is in error. I am a volunteer math coach to some junior college students and can't find my error in the problem. Its been a while since I earned my masters in electrical engineering.

The issue is the presencce of the variable "a" in the denominator of two of the terms in the answer. I can't imagine why the "a"s are not in the correct answer.

Homework Equations

The problem requires solution by trigonometric substitution. All other equations are listed in my attempt at the problem.

The Attempt at a Solution

My work is all on the solution attempt.

 

[WWGD]

Seems like an $x=atan \theta$ or $x=asec \theta$ would work.

 

[Ray Beaver]

Seems like an $x=atan \theta$ or $x=asec \theta$ would work.

The solution I provided attempts to use x = a tan theta but the solution is incorrect

 

[WWGD]

The solution I provided attempts to use x = a tan theta but the solution is incorrect

Is the solution an indefinite (general) or definite integral (numerical value)?

EDIT: Nevermind, I did not read carefully and did not see the 3/2 exponent. Please ignore and let me
rethink. It is a variant of the one with power 1/2, but needs some changes.

 

[Ray Beaver]

The solution is listed at the bottom of the attempt I posted and is indefinite.

 

[Ray Beaver]

The solution at the bottom of the solution page is missing a minus sign in front of the first term, my mistake in listing the correct answer. My answer has that term correct, but is in error in the argument of the natural log.