evaluate ∫2/[(1-x)(1+x 2 ] dx


∫2/[(1-x)(1+x2] dx

Grade:12th Pass

1 Answers

Rinkoo Gupta
askIITians Faculty 80 Points
8 years ago
integral 2/[(1-x)(1+x^2)]dx
=2 integral dx/(1-x)(1+x^2)
Using Partial fraction we decompose the function
1/(1-x)(1+x^2)=A/(1-x) +(Bx+C)/(1+x^2)
on simplifying we get A=1/2,B=1/2 and C=1/2
So 1/(1-x)(1+x^2)=1/2(1-x) +1/2(x+1)/(1+x^2)
=>2[ integral[1/(2(1-x)) +1/2 (x/(1+x^2) +1/(1+x^2))]]dx
=2[-(1/2)logmod(1-x)+1/2integral(x/(1+x^2))dx +1/2 .inverse of tanx] +c
=2[-(1/2)logmod(1-x) +1/4log(1+x^2) +1/2. inverse of tanx ]+C
Thanks & Regards
Rinkoo Gupta
AskIITians Faculty

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