How to find the general antiderivative of f ( x ) = x ( 6 &#x2212;<!-- − --> x

Kapalci

Kapalci

Answered question

2022-06-27

How to find the general antiderivative of f ( x ) = x ( 6 x ) 2 ?
I want to find the general antiderivative of f ( x ) = x ( 6 x ) 2 . However, I keep getting it wrong.
I am new to antiderivatives, but I think the first thing I should do is differentiate.
d d x ( 6 x ) 2 = 2 ( 6 x ) 1 = 2 ( 6 x )
f ( x ) = [ ( 6 x ) 2 1 ] + [ 2 ( 6 x ) x ] = ( 6 x ) 2 2 x ( 6 x )
According to the antiderivative power rule, when n 1, x n d x = x n + 1 n + 1 + C.
So it seems like ( 6 x ) 2 d x = ( 6 x ) 3 3 + C.
However, I can't find a rule that seems like it would work with 2 x ( 6 x ). The closest thing that I can find is the "Multiplication by Constant Rule", c f ( x ) d x = c ( f ( x ) ) d x, but I'm not sure if I'm allowed to change 2 x ( 6 x ) into the form 2 ( 6 x x 2 ).
2 ( 6 x x 2 )
= 2 ( 6 x x 2 )
= 2 ( 6 x x 2 )
= 2 ( 6 x 2 2 x 3 3 ) + C
But ( 6 x ) 3 3 ( 6 x 2 2 x 3 3 ) + C is incorrect.

Answer & Explanation

assumintdz

assumintdz

Beginner2022-06-28Added 22 answers

An alternative strategy is to substitute y = x  6, so
f ( x ) = 6 y 2 + y 3 = d d y ( 2 y 3 + 1 4 y 4 ) = d d x y 3 ( y + 8 ) 4 = d d x ( x  6 ) 3 ( x + 2 ) 4 .

Hence, the basic antiderivative is ( x  6 ) 3 ( x + 2 ) 4 + C.

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