Answer:
see below
Step-by-step explanation:
<h2>Part-A:</h2>
we want to find the quotient and remainder when 4x²+4x is divided by 2x+1 in other words we want to find the quotient and remainder when:
To do so, I would prefer using simple algebra rather than using troublesome polynomial long division. anyway dividing it would yield the form where:
- p(x) is the quotation
- k is the remainder
- q(x) is the divisor
Therefore,In order to derive the quotation and remainder, rewrite the numerator which yields:
Compering it to the mentioned form, we can consider:
- 2x+1, The quotient
- -1, The remainder
<h2>Part-B:</h2>
we are asked to show that,
Well,we can start with integrating the indefinite integral and the first step to do so is to decompose the fraction, integrand. As we have already done it in part-a, we can simply skip the steps:
utilizing sum integration rule yields:
apply constant integration rule which yields:
recall that,
- integration of xⁿ is xⁿ+¹/n+1
- integration of a constant,k is kx
so we derive from utilizing the rules is that,
Now integrating would require <u>u-substitution </u> . In order to perform the substitution, let
To perform the substitution multiply the integrand and integral by 2 and ½ respectively:
perform the substitution:
integrating yields:
back-substitute:
To evaluate the define integral , return the limits of integration:
remember fundamental theorem
utilize it and simplify which yields:
and we are done!