integrate 4ax+2b/(ax^2 +bx+c)^10 using substitution

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Solution

let us consider "t" as "(ax² + bx + c)"

t = (ax² + bx + c)

differentiate with respect to x

dt = (2 a x + b (1) + 0) dx

dt = (2 a x + b) dx

now we are going to apply the value of t and dt in the given question

= ∫(4ax + 2b)/(ax² + bx + c)^10 dx

now we are going to take 2 from the numerator

= ∫ 2 (2 ax + 2b)/(ax² + bx + c)^10 dx

= ∫ 2 (dt/t^10)

= ∫ 2 t^-10 dt

= 2 t^(-10 + 1)/(-10 + 1) + C

= 2 t^(-9)/(-9) + C

= (-2/9) (ax² + bx + c)^(-9) + C

= [-2/9(ax² + bx + c)^9] + C

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