int(4dx)/((x^(2)-2x+5)^(2))

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Identify Integral: Identify the integral to be solved. We need to evaluate the integral of the function (4dx)/((x^2 - 2x + 5)^2).Complete Square: Complete the square for the denominator. The denominator is a quadratic expression, and we can complete the square to make it easier to integrate. The expression x^2 - 2x + 5 can be written as (x - 1)^2 + 4, since (x - 1)^2 = x^2 - 2x + 1 and we add 4 to get back to 5.Rewrite Integral: Rewrite the integral with the completed square. The integral now becomes int(4dx)/(((x - 1)^2 + 4)^2).Use Substitution: Use substitution to simplify the integral. Let u = x - 1, which implies du = dx. The limits of integration do not change because they are infinite. The integral now becomes int(4du)/((u^2 + 4)^2).Recognize Standard Form: Recognize the integral as a standard form. The integral int(4du)/((u^2 + 4)^2) is a standard integral that can be solved using a table of integrals or integration techniques such as partial fractions or trigonometric substitution. However, in this case, we can use a direct substitution because the denominator is a perfect square of a binomial.Trigonometric Substitution: Use a trigonometric substitution. Let u = 2tan(θ), which implies du = 2sec^2(θ)dθ. The integral becomes int(4 * 2sec^2(θ)dθ)/((2tan(θ)^2 + 4)^2). Simplify the denominator: (2tan(θ)^2 + 4) = 4(tan(θ)^2 + 1) = 4sec^2(θ).Substitute and Simplify: Substitute and simplify the integral. The integral now becomes int(8sec^2(θ)dθ)/(16sec^4(θ)). This simplifies to int(1/2sec^2(θ)dθ)/(sec^4(θ)) = int(1/2)dθ/(sec^2(θ)).Simplify Using Identities: Simplify the integral using trigonometric identities. We know that sec^2(θ) = 1/cos^2(θ), so the integral becomes int(1/2)dθ/(1/cos^2(θ)) = int(1/2)cos^2(θ)dθ.Integrate Using Identities: Use a trigonometric identity to integrate. The trigonometric identity cos^2(θ) = (1 + cos(2θ))/2 can be used to integrate. The integral becomes int(1/2)(1 + cos(2θ))/2dθ = int(1/4)dθ + int(1/4)cos(2θ)dθ.Convert to Original Variable: Integrate using basic trigonometric integrals. The integral of int(1/4)dθ is (1/4)θ, and the integral of int(1/4)cos(2θ)dθ is (1/4)(1/2)sin(2θ) = (1/8)sin(2θ). So the integral becomes (1/4)θ + (1/8)sin(2θ) + C, where C is the constant of integration.Express Theta in Terms: Convert back to the original variable x. We need to express θ and sin(2θ) in terms of x. From u = 2tan(θ), we have tan(θ) = u/2 = (x - 1)/2. We can draw a right triangle with the opposite side as (x - 1)/2, the adjacent side as 2, and the hypotenuse as √((x - 1)^2/4 + 4). Using the definition of tangent and sine, we can find θ and sin(2θ) in terms of x.Express Sin(2Theta) in Terms: Express θ in terms of x. Using the arctangent function, we have θ = arctan((x - 1)/2).Express sin(2θ) in terms of x. Using the double-angle formula for sine, sin(2θ) = 2sin(θ)cos(θ). We can find sin(θ) and cos(θ) from the triangle we constructed. However, this step involves a lot of algebraic manipulation and is prone to errors. Instead, we can recognize that this integral is better solved using a different method, such as partial fractions, which avoids trigonometric substitution altogether. This realization indicates that we have made a mistake in choosing the method of integration.

$\int \frac{4\,dx}{(x^{2}-2x+5)^{2}}$

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$\int \frac{4\,dx}{(x^{2}-2x+5)^{2}}$