int((x^(3)-2x^(2)+1)dx)/((x^(2)-2x+5)^(2))

Question

Bytelearn · Accepted Answer

Given Integral: We are given the integral: int((x^3 - 2x^2 + 1)dx) / ((x^2 - 2x + 5)^2) To solve this integral, we will use the method of partial fractions. However, the denominator (x^2 - 2x + 5)^2 does not factor into linear terms over the real numbers because the discriminant (2^2 - 4*1*5) is negative. Therefore, we cannot decompose the fraction into simpler partial fractions directly. Instead, we will complete the square for the denominator and then try to simplify the integral.Complete the Square: First, let's complete the square for the denominator: x^2 - 2x + 5 = (x^2 - 2x + 1) + 4 = (x - 1)^2 + 4 Now, the integral becomes: int((x^3 - 2x^2 + 1)dx) / ((x - 1)^2 + 4)^2Substitution: Next, we will make a substitution to simplify the integral. Let: u = x - 1 Then, du = dx, and x = u + 1 Now we can rewrite the integral in terms of u: int((u + 1)^3 - 2(u + 1)^2 + 1)du / (u^2 + 4)^2Expand and Simplify: We expand the numerator to simplify the expression: int((u^3 + 3u^2 + 3u + 1) - 2(u^2 + 2u + 1) + 1)du / (u^2 + 4)^2 = int((u^3 + 3u^2 + 3u + 1 - 2u^2 - 4u - 2 + 1)du) / (u^2 + 4)^2 = int((u^3 + u^2 - u)du) / (u^2 + 4)^2Split into Integrals: Now, we will split the integral into three separate integrals: int(u^3 du) / (u^2 + 4)^2 + int(u^2 du) / (u^2 + 4)^2 - int(u du) / (u^2 + 4)^2Complex Integral: The first integral int(u^3 du) / (u^2 + 4)^2 is a bit complex and does not have a straightforward antiderivative. We will need to use a method such as integration by parts or a further substitution. However, this is a non-standard integral and may require a special function or a numerical method to solve. At this point, we realize that the approach we have taken may not be the most efficient, and we may need to reconsider our strategy.

$\int\frac{x^{3}-2x^{2}+1}{(x^{2}-2x+5)^{2}}dx$

Full solution

More problems from Find indefinite integrals using the substitution

Related topics

∫x3−2x2+1(x2−2x+5)2dx\int\frac{x^{3}-2x^{2}+1}{(x^{2}-2x+5)^{2}}dx∫(x2−2x+5)2x3−2x2+1​dx

$\int\frac{x^{3}-2x^{2}+1}{(x^{2}-2x+5)^{2}}dx$