Calculating integral by differentiating first

[physmatics]

Homework Statement

Calculate the integral
$$I = \int (t^x - 1)/ln(t) dt, boundaries: 0 \leq t \leq 1, x \geq 0$$
by differentiating first with respect to x.

Homework Equations

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The Attempt at a Solution

I have no idea how to solve this, but it's on our sample exam and there are no solutions... =/ Differentiating with respect to x gives me:
$$d/dx((t^x - 1)/ln(t)) = ln(t)e^(xln(t))/ln(t) + 0 = e^(xln(t)) = t^x$$
Can I use this in any way? Maybe substitute t^x in the integral with d/dx(t^x - 1)/ln(t)?
Any clues appreciated!

 

[micromass]

Remember that the Leibniz Rule gives you

$$\frac{d}{dx}\int_0^1{\frac{t^x-1}{ln(t)}dt}=\int_0^1{\frac{d}{dx}\left(\frac{t^x-1}{ln(t)}\right)dt}=\int_0^1{t^xdt}$$

Now, the right-hand side looks like an easy integral...

 

[HallsofIvy]

Very nice. Physmatics, becareful to distinguish between the derivatives of $x^a$ and $a^x$.