Determine the necessary or sufficient conditions for the ∫P(x)/q(x) dx Be an immediate integral, where p(x) is a grade 2 polynomial and q(x) a grade 3 polynomial Considerations 1. Let g be a Real root of q(x). Then q(x)=(x-g) q1(x), where q1(x) is a polynomial of degree 2. 2. Find constants s,y,m so that p(x)=ax2+bx+c = s(q'(x)) + y(x-g) + m 3. Without loss of generality we can consider q(x)= fx3 +hx2 +jx +k In the end, this means that ∫ p(x)/q(x)dx=s∫ q'(x)/q(x)dx+y∫(x-g)/q(x)dx+∫ (m)/q(x) = s∫q'(x)/q(x)dx+y∫dx/q1(x) + m∫dx/q(x). Depending on whether q1(x) is an irreducible quadratic polynomial or not, it can be seen that the above can easily be expressed as s ln (q(x)) + y ∫ dx/q1(x) + m∫dx/q(x)
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Determine the necessary or sufficient conditions for the ∫P(x)/q(x) dx
Be an immediate integral, where p(x) is a grade 2 polynomial and q(x) a grade 3 polynomial
Considerations
1. Let g be a Real root of q(x). Then q(x)=(x-g) q1(x), where q1(x) is a polynomial of degree 2.
2. Find constants s,y,m so that p(x)=ax2+bx+c = s(q'(x)) + y(x-g) + m
3. Without loss of generality we can consider q(x)= fx3 +hx2 +jx +k
In the end, this means that
∫ p(x)/q(x)dx=s∫ q'(x)/q(x)dx+y∫(x-g)/q(x)dx+∫ (m)/q(x)
= s∫q'(x)/q(x)dx+y∫dx/q1(x) + m∫dx/q(x).
Depending on whether q1(x) is an irreducible quadratic polynomial or not, it can be seen that the above can easily be expressed as
s ln (q(x)) + y ∫ dx/q1(x) + m∫dx/q(x)
Note: Don't use the partial fraction method to integrate and a numerical example
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