00:01
This problem, we can see that the denominator factors into x times 7x squared plus 1, right? meaning our partial fractions are a over x plus bx plus c over 7x squared plus 1, right? and this all equals the numerator right here, however, because we have to get a common name denominator, we need to multiply up with sides, right? which leads us to have a multiplied by 7x squared plus 1 plus bx plus c times x right and this becomes 7 a x squared plus a plus bx squared plus cx equals 9 minus x squared right so we're to gather all the x squared values we would have 7a x squared plus bx squared plus cx equals 9 minus x squared right so we're together all the x squared values we would have 7a plus b equals negative 1 and and a equals 9, right? also, c equals 0, right? because there is no value of x available.
01:09
So in that case, we can ignore the c here, right? and then because a equals 9, we get that b equals negative 64, right? so we can directly plug a into this, into this partial fraction right here, and integrate.
01:26
So that would be 9 over x and 0 .9 over x dx.
01:31
It would get this as 9 times ln of x plus c.
01:36
However, now we have to focus on this, right? and this becomes negative 64x over 7x squared plus 1 dx.
01:45
This is the integral that we're left with on that side...