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Question

Integral x​​​​​​0 to x (dx) = integral 0 to t ( v dt )

= integral 0 to t ( v​​​​​​0 +at ) dt x-x0 =v​​​​​​0 t + 1/2 at​​​​​2 . This is my confused part

How we reached🔼 . Please explain

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Solution

Integral x​​​​​​0 to x (dx) =
x0-x

I hope this step is clear to you. Simple integration.

Now,
We know V =U+at =V0+at
This is the velocity relation.
We substituted this for V.
Then integrated that.
Integral (V0+at) = V0t +(1/2)at²

Please note that the RHS is integrated with respect to t.

Hope it clears.
Good luck.

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