How to find the general antiderivative of f ( x ) = x ( 6...

Kapalci

Kapalci

Answered

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

Expert

2022-06-28Added 22 answers

Step 1
Another way to do it 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 .
Step 2
Thus the general antiderivative is ( x 6 ) 3 ( x + 2 ) 4 + C.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?