Leibniz integral rule:
dxd(∫a(x)b(x)f(x,t)dt)=f(x,b(x))×dxdb(x)−f(x,a(x))×dxda(x)+∫a(x)b(x)∂x∂f(x,t)dt
In your case :
dxd∫1xtan2(t2)dt=?
Then:
a(x)=1
b(x)=x
f(x,t)=tan2(t2)
Take derivatives with respect to x:
dxda(x)=dxd1=0
dxdb(x)=dxdx=2x1
∂x∂f(x,t)=∂x∂tan2(t2)=0
(comment: in this case f(x,t) is independent of the variable x, so partial derivative with respect to t equals zero).
f(x,b(x)) means f(x,t) with a fixed value of a variable t, t takes on a value equal to b(x).
Eventually we have:
dxd∫1xtan2(t2)dt=2xtan2(x)−tan2(12)×0+∫1x0dt=2xtan2(x)
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