00:01
In this problem, we have to perform indefinite integration of six different questions.
00:08
So first, we have to integrate x -cube minus 2x whole square into 3x square minus 2 -dx.
00:23
Let us assume that x -cube minus 2x is equal to t.
00:29
So we will perform this integration by integration by the method of substitution.
00:34
So let us assume that x cube minus 2x is equals to t.
00:40
So if we differentiate this entire function, so what will be the differentiation of left hand side? so this will be 3x square minus 2 dx that will be equals to d t.
00:55
So in above expression we see that this is the t and this entire term is nothing but d t.
01:02
Therefore, the above integration will reduce to integration of t squared into dt.
01:10
Now integration of t square will be t -rase to the power 2 plus 1 divided by 2 plus 1 plus 1 plus there will be an integration constant since this is an indefinite integration.
01:23
So this will be equals to t -cube divided by 3 plus constant.
01:31
Now we can replace the t with the substitution that we had done earlier.
01:37
So if you replace the t by the substitution, so this equation will be x cube minus 2x divided by 3 and the numerator in numerator there will be whole cube plus integration constant.
01:52
Therefore integration of x cube minus 2x whole square into 3x square.
02:02
Where minus 2 dx will be equals to x cube minus 2x whole cube upon 3 plus c where c is an integration constant so this is the answer for first problem okay now moving to the second problem second question is integration of 2 x -raise to the power 3 by 4 into d x now the cost constant can be taken outside the integration.
02:52
So this integration will be equals to 2 times integration of x -raise to the power 3x dx, which is equal to 2 times.
03:02
Now integration of x -res -to -the -power n is x -res -to -the -power n plus 1.
03:07
So here n is 3 by 4, so we will add 1 in the power and divide by the same thing.
03:13
And there will be an integration constant c.
03:16
So this will be equal to a denominator will be 7 by 4.
03:21
So if we reverse this, this is going to be 4 upon 7 into x -res to the power 7 by 4 plus constant, which is equal to 8 by 7...