Antiderivative of ( x 2 </msup> + c ) <mrow class="MJX-TeXA

Aedan Tyler

Aedan Tyler

Answered question

2022-05-08

Antiderivative of ( x 2 + c ) 3 / 2
What method should be used to determine the antiderivative of this expression?
Edit: I have c > 0 in the problem I'm working on.

Answer & Explanation

Duncan Cox

Duncan Cox

Beginner2022-05-09Added 18 answers

Step 1
( x 2 + c ) 3 / 2 d x = 1 ( x 2 + c ) 3
Let us use a new variable t such that x 2 + c = x + t. This is the first Euler substitution.
From this formula we can solve for x to get x = c t 2 2 t , d x = c t 2 2 t 2 d t, x 2 + c = x + t = c + t 2 2 t , and ( x 2 + c ) 3 = ( c + t 2 ) 3 8 t 3 .
Step 2
We put these into the integral and get
8 t 3 ( c + t 2 ) 3 ( c + t 2 ) 2 t 2 d t = 2 2 t ( c + t 2 ) 2 d t = 2 1 c + t 2
We can return to the variable x since t = x 2 + c x.
Therefore, ( x 2 + c ) 3 / 2 = 2 c + ( x 2 + c x ) 2

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