Did the author make a mistake in integrating by parts?

[bobsmith76]

Homework Statement

In this video from which there is a screen shot above the author went from x/2 to 2x and all he said was half is two quarters. right a half is two quarters it is not 2. I just want to make sure that he made a mistake because I've been seeing some real bizarre things in the solution manual re integration by parts that i do not understand. everything else in the above simplification i understand except how

x/2 cos becomes 2x cos

 

[Sourabh N]

Notice the $\frac{1}{4}$ outside the bracket? $\frac{x}{2} cos$ becomes $\frac{1}{4}(2x*cos$

 

[bobsmith76]

I would think 1/4 * x/2 would make x/8.

 

[Sourabh N]

I would think 1/4 * x/2 would make x/8.

Indeed. Look at what I wrote, again. Let me rephrase it: $\frac{x}{2} cos$ = $\frac{1}{4}(2x*cos$

 

[bobsmith76]

are you saying somehow they got (x/2)/(1/4). That would make 1/2 * 4, but I don't see how that move is justified.

 

[Sourabh N]

Indeed. Look at what I wrote, again. Let me rephrase it: $\frac{x}{2} cos$ = $\frac{1}{4}(2x*cos$

are you saying somehow they got (x/2)/(1/4). That would make 1/2 * 4, but I don't see how that move is justified.

Nope! In the first step they have $\frac{x}{2}$. right?
In the second step, they have $\frac{1}{4}2x$. right?
Now, $\frac{x}{2}$ = $\frac{1}{4}2x$. right?

 

[bobsmith76]

Ok, I got it. Thanks. Sorry for being so hard-headed

 

[Sourabh N]

Happens You're welcome!