00:01
For this problem we are to evaluate the definite integral from 0 to 1 of sine of x -rays to the 4th power dx using pair of series, and we will approximate it correct to four decimal places.
00:14
Now we recall, sine of x has a power series representation of summation from n equals 0 to infinity of negative 1 raise to n times x raised 2n plus 1, all over 2n plus 1 factorial.
00:32
So, sign of x -rays to the 4th power is equal to summation from n equals 0 to infinity, negative 1 raised to n times x -rays to the 4th power raised to 2n plus 1, all over 2n plus 1 factorial.
00:50
Now, if we are to expand the power series of sine of x, that's x minus x cubed over 3 factorial, plus x raised to the fifth over 5 factorial minus x raise to the seventh power over 7 factorial and so on, which means that if we are to expand this power series for sign of x raised to the 4th, that's x raised to the 4th minus x to the 4th raise to the 3 power over 3 factorial, plus x to the 4th raise to the 5th power over 5 factorial, minus x to the fourth power raised to the seventh power over seven factorial and so on.
01:37
That's the same as x to the fourth minus x raised the power of 12 over 3 factorial plus x raised to 20 over 5 factorial minus x raised to 28 over 7 factorial and so on.
01:52
Thus, antiderivative from 0 to 1 of sine of x -rays to the 4th dx is equal to integral from 0 to 1 of this summation, which is the same as x -rays to the 4th minus x -rays to the 12 over 3 factorial, x -rays to 20 over 5 factorial, minus x -rays to 28 over 7 factorial, and so on.
02:24
And then d x...