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:

"\\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 :

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


Then:

"a(x)=1"

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

"f(x,t)=tan^2(t^2)"


Take derivatives with respect to x:

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

"\\frac{d}{dx}b(x)=\\frac{d}{dx}\\sqrt{x} = \\frac{1}{2\\sqrt{x}}"

"\\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))" means f(x,t) with a fixed value of a variable t, t takes on a value equal to b(x).


Eventually we have:

"\\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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