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Integral Calculus

∫dx/(x^2-2x+4)^3/2 please provide me detailed solution of this step wise solution . it defenite integral where lower limit is 1 and upper limit is 2

Question image for ∫dx/(x^2-2x+4)^3/2 please provide me detailed sol

Amit Singh

7 Years agoGrade

1 Answer

Vinod Ramakrishnan Eswaran

7 Years ago

$I=\int_{1}^{2}dx/(x^{2}-2x+4)^{3/2}$

$I=\int_{1}^{2}dx/(x^{2}-2x+4)^{3/2}\; =>\int_{1}^{2}dx/((x-1)^{2}+3)^{3/2}$

$Substitute\: x-1\: as\: t,\: we\: get\: dx=dt$

$I=\int_{0}^{1}dt/(t^{2}+3)^{3/2}$

$We\: can\: use\: formula\: I=\int dx/(a^{2}+x^{2})^{3/2}=x/((a^{2})(a^{2}+x^{2})^{1/2})$ *

$We\: get\: I=t/(3(3+t^{2})^{1/2})^{1}_{0}\:$

Solving we get I=1/6

Hence this is the solution

For the proof for the * we need to substitute x as atanx and then integrate

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