Answer to Question #102231 in Calculus for DEBASIS BEHERA

Question #102231
d/dx int 1 to rootx tan^2(t^2)dt
1
Expert's answer
2020-02-04T09:04:32-0500

Leibniz integral rule:

ddx(a(x)b(x)f(x,t)dt)=f(x,b(x))×ddxb(x)f(x,a(x))×ddxa(x)+a(x)b(x)xf(x,t)dt\frac{d}{dx}(\int_{a(x)}^{b(x)} f(x,t)dt)= f(x,b(x))\times \frac{d}{dx}b(x)-f(x,a(x))\times\frac{d}{dx}a(x) +\int_{a(x)}^{b(x)}\frac{\partial}{\partial x}f(x,t)dt

In your case :

ddx1xtan2(t2)dt=?\frac{d}{dx} \int_{1}^{\sqrt{x}} tan^2(t^2)dt= ?


Then:

a(x)=1a(x)=1

b(x)=xb(x) = \sqrt{x}

f(x,t)=tan2(t2)f(x,t)=tan^2(t^2)


Take derivatives with respect to x:

ddxa(x)=ddx1=0\frac{d}{dx}a(x) = \frac{d}{dx}1 = 0

ddxb(x)=ddxx=12x\frac{d}{dx}b(x)=\frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}

xf(x,t)=xtan2(t2)=0\frac{\partial }{\partial x}f(x,t) = \frac{\partial }{\partial x}tan^2(t^2)=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))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:

ddx1xtan2(t2)dt=tan2(x)2xtan2(12)×0+1x0dt=tan2(x)2x\frac{d}{dx} \int_{1}^{\sqrt{x}} tan^2(t^2)dt=\frac{ tan^2(x)}{2\sqrt{x}}- tan^2(1^2)\times 0 +\int_{1}^{\sqrt{x}}0dt= \frac{ tan^2(x)}{2\sqrt{x}}


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