=int((16 x-20)dx)/(4x^(2)-12 x+10)

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Given Integral Simplification: We are given the integral: ∫((16x - 20)dx) / (4x^2 - 12x + 10) First, we should try to simplify the denominator if possible and see if we can factor it. The denominator is a quadratic expression, so we can attempt to factor it by completing the square or using the quadratic formula.Completing the Square: The quadratic expression in the denominator is: 4x^2 - 12x + 10 To complete the square, we can factor out the 4 from the first two terms to get: 4(x^2 - 3x) + 10 Now, we need to add and subtract the square of half the coefficient of x inside the parentheses, which is (3/2)^2 = 9/4. 4(x^2 - 3x + 9/4 - 9/4) + 10 4(x - 3/2)^2 - 4(9/4) + 10 4(x - 3/2)^2 - 9 + 10 4(x - 3/2)^2 + 1 Now we have the denominator in the form of a perfect square plus a constant.Substitution for Simplification: Next, we can rewrite the integral with the simplified denominator: ∫((16x - 20)dx) / (4(x - 3/2)^2 + 1) Now, we should look for a substitution that will simplify the integral. A good substitution would be to let u equal the expression inside the square, i.e., u = x - 3/2.Performing Substitution: Let's perform the substitution: u = x - 3/2 Then, du = dx, since the derivative of x with respect to x is 1 and the derivative of a constant is 0. Now we need to express the numerator (16x - 20) in terms of u. To do this, we solve for x in terms of u: x = u + 3/2Splitting into Two Integrals: Substitute x in the numerator: 16x - 20 = 16(u + 3/2) - 20 = 16u + 24 - 20 = 16u + 4 Now we can rewrite the integral in terms of u: ∫((16u + 4)du) / (4u^2 + 1)First Integral Simplification: We can split the integral into two separate integrals: ∫(16u du) / (4u^2 + 1) + ∫(4 du) / (4u^2 + 1) Now, we can simplify each integral further. For the first integral, we can factor out constants and for the second integral, we can use a standard integral formula.Second Integral Simplification: The first integral becomes: 4∫(4u du) / (4u^2 + 1) We can factor out the 4 from the numerator and cancel it with the 4 in the denominator: ∫(4u du) / (4u^2 + 1) = ∫(u du) / (u^2 + (1/4))Integrating Both Parts: The second integral is a standard form: ∫(4 du) / (4u^2 + 1) We can factor out the 4 from the denominator: ∫(4 du) / (4(u^2 + (1/4))) = ∫(du) / (u^2 + (1/4)) This is the integral of the form ∫(du) / (u^2 + a^2), which has a standard result of (1/a)arctan(u/a) + C.Final Result Substitution: Now we integrate both parts: For the first integral, we have: ∫(u du) / (u^2 + (1/4)) = (1/2)ln|u^2 + (1/4)| + C1 For the second integral, we have: ∫(du) / (u^2 + (1/4)) = 2arctan(2u) + C2Combining both integrals, we get the final result: (1/2)ln|u^2 + (1/4)| + 2arctan(2u) + C Now we need to substitute back for u to get the result in terms of x: u = x - 3/2Substituting back, we get: (1/2)ln|(x - 3/2)^2 + (1/4)| + 2arctan(2(x - 3/2)) + C This is the indefinite integral of the given function.

$\int \frac{(16x-20)\,dx}{4x^{2}-12x+10}$

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∫(16x−20) dx4x2−12x+10\int \frac{(16x-20)\,dx}{4x^{2}-12x+10}∫4x2−12x+10(16x−20)dx​

$\int \frac{(16x-20)\,dx}{4x^{2}-12x+10}$