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(Solved): Consider the expression (x^(3)+5x^(2)+4)/(x^(2)+4). The numerator is a polynomial of degree 2, and ...
Consider the expression
(x^(3)+5x^(2)+4)/(x^(2)+4). The numerator is a polynomial of degree 2, and the denominator is an irreducible quadratic. Because the degree of the numerator is greater than the degree of the denominator, we can use long division to find real numbers
a,b,c,dto write
(x^(3)+5x^(2)+4)/(x^(2)+4)= polynomial + remainder
=(ax+b)+(cx+d)/(x^(2)+4)Using long division:
(x^(3)+5x^(2)+4)/(x^(2)+4)=
x+,(x)/(x^(2)+4)+,(1)/(x^(2)+4)Therefore
\int (x^(3)+5x^(2)+4)/(x^(2)+4)dx=(Hint: Write expressions in
xusing the same order as above!
