00:01
In the first question, we want to integrate x cubed plus 2x over x to the power of 4 plus 4x squared plus 7.
00:17
Now i want to use the form of f prime over fx.
00:23
If we can fit this form, then it will just be ln mod of fx plus c.
00:31
So if i were to let my fx be the denominator, f prime would be differentiating x to the power of 4, bring down the power of 4, repeat the x and subtract 1 to the power, we get 3.
00:50
4 is multiplied by x squared so it's kept aside.
00:54
Differentiating x squared, bring down the power of 2, repeat the x and subtract 1 to the power, so it becomes power of 1.
01:01
7 is a constant, we differentiate, we get 0.
01:03
So f prime is 4x cubed plus 8x.
01:10
Factorise out the 4, i'll get x cubed plus 2x.
01:14
You can see that it's over here but lacking a 4.
01:18
So i'm going to multiply a 4 to the numerator, so i'm going to multiply a quarter in front to balance.
01:33
You can see that this portion here is f prime x and this portion here is fx.
01:43
So it fitted this form and hence i can use the results.
01:48
And so that will be a quarter ln or mod the denominator.
01:57
This is always going to be greater than 0 because x to the power of 4 is greater than or equal to 0.
02:04
X squared is greater than or equal to 0 plus a 7, definitely greater than 0.
02:08
So i can remove the mod.
02:10
So it'll be a quarter ln, just use the round bracket.
02:21
Next, i want to integrate 1 plus 3 ln x squared over x dx.
02:31
Now i'm going to use the form of f prime multiplied to fx to the power of n.
02:43
If we can fit this form, then we can use the result fx to the power of n plus 1 to the power divided by the new power plus c.
02:54
So now i'm going to let fx be 1 plus 3 ln x.
03:03
So f prime will be, differentiating 1 which is a constant, you get 0.
03:10
3 is multiplied variable so it's kept aside.
03:12
Differentiating ln x, i'll get 1 over x.
03:17
So f prime x is 3 over x...