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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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