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Question

The value of the integral 10xα1logxdx is

A
log(α+1)
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B
2log(α+1)
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C
3logα
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D
none of these
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Solution

The correct option is A log(α+1)
I(α)=10xα1log xdx

Differentiating with respect to α, we get,
I(α)=10xαdx=1α+1
We need to solve this simple differential equation :I(α)=1α+1
On solving, we get,
I(α)=log(α+1)+C

Now, from the initial equation, we can find that I(0)=0
I(0)=log(1)+C=0 C=0

So, I(α)=log(α+1)

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