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# Evaluate ∫ 1 0 X a − 1 Log X D X - Applied Mathematics 2

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Evaluate int_0^1( x^a-1)/log x dx

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#### Solution

let int_0^1( x^a-1)/log x dx

Taking ‘a’ as parameter ,

I (a)= int_0^1(x^a-1)/log x dx        -------- (1)

differentiate w.r.t a ,

(dI(a))/(da)=d/(da) int_0^1 (x^a-1)/log x dx

∴(dI(a))/(da)=int_0^1 del/del_a (x^a-1)/log x dx ………{ D.U.I.S f(x)}

∴(dI(a))/(da)= int_0^1 (x^a log x)/log x    dx ……… {(dx^a)/(da)=x^a. log a} v

∴( dI(a))/(da)= int_0^1 x^a  dx

∴ (dI (a))/(da)=[(x^a+1)/(a+1)]1/0

∴ (dI(a))/(da)=1/(a+1)-0

∴ (dI(a))/(da)=1/(a+1)

now , integrate w.r.t a, I (a)= int 1/(a+1) da

I(a)= log(a+1)+c            -------- (2)

where c is constant of integration
put a=0 in eqn (1),

I(0)=int_0^1 0  dx=0

And
From eqn (2), I (0)=c

∴ c=0

∴ I= log (a+1)

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
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