00:01
In this question, we have to calculate what is the convolution of 2 times t star t raised to 4.
00:11
Here star represents the convolution.
00:14
So, first of all, let me write down the general formula for it.
00:19
We know that f star g of t the convolution of f and g is given by integral 0 to t f of u times g of t minus u du correct so here we can take this as let's write down here we are taking f of t as 2 times of t and g of t as t raised to 4 so this comes out to be integral 0 to t f of u it will be 2 times u and g of t minus u which comes out to be t minus u raised to 4 du so what we can do we can simply find this at 0 to t and here what we are having 2 we can take out u times t minus u raised to 4.
01:18
This we can write it as t minus u whole square.
01:23
So this is going to be t square plus u square minus 2u t whole square du.
01:36
Again, we have to expand integral 0 to t u times of, this is a plus b plus c whole square, correct? so what we are going to get? a square plus b square plus c square correct plus 2 ab so 2t square u square plus 2bc so what we are going to get minus 4 u cube t minus 2ac if we do we are going to get 4 ut cube this is what we we have.
02:13
So, this now we can multiply t inside we have u t raised to 4 plus u raised to 5 plus 4 u cube t square plus 2 t square u cube minus 4 u raised to 4 t minus 4 u square t cube.
02:34
This is what we are getting du.
02:38
Just now we can integrate simply...