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

Question

Bytelearn · Accepted Answer

Simplify Denominator: Simplify the denominator of the integrand. We have the integral: int((16x - 20)dx) / (4x^2 - 12x + 10) First, we can factor out a 4 from the denominator to simplify the expression: int((16x - 20)dx) / (4(x^2 - 3x + 2.5)) Now, we can cancel out the common factor of 4 from the numerator and the denominator: int((4(4x - 5)dx) / (4(x^2 - 3x + 2.5))) = int((4x - 5)dx) / (x^2 - 3x + 2.5)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: x^2 - 3x + 2.5 can be written as (x - 1.5)^2 - 1.5^2 + 2.5 = (x - 1.5)^2 - 0.25 Now, the integral becomes: int((4x - 5)dx) / ((x - 1.5)^2 - 0.25)Use Substitution: Use substitution to solve the integral. Let u = (x - 1.5)^2 - 0.25, then du = 2(x - 1.5)dx. To match the numerator (4x - 5), we need to express it in terms of (x - 1.5). We can rewrite the numerator as: 4x - 5 = 4(x - 1.5) + 1 Now, we can split the integral into two parts: int((4(x - 1.5) + 1)dx) / (u) = 4 * int((x - 1.5)dx) / (u) + int(dx) / (u) For the first part, we can use the substitution: 4 * int((x - 1.5)dx) / (u) = 4 * 1/2 * int(du) / (u) = 2 * int(du) / (u) For the second part, we need to adjust the substitution to match the dx term: int(dx) / (u) = int(du) / (2(x - 1.5)(u)) However, we cannot directly integrate the second part as it stands because it does not match our substitution. We need to express (x - 1.5) in terms of u.

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

Full solution

More problems from Find indefinite integrals using the substitution

Related topics

∫(16x−204x2−12x+10)dx\int\left(\frac{16x-20}{4x^{2}-12x+10}\right)dx∫(4x2−12x+1016x−20​)dx

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