61–68. Derivatives of integrals Simplify the following expressions. 61. + 1 + 1) dt d 62. dx e' dt -10 d 64. dx dp dz 63. dx z2 + 1 VA + 1 dt dp 65. 66. dx p² + 1 d 67. dx Vi + t² dt In 1? dt 68. dx

Q: 1. INTEGRALS AND ITS APPLICATIONS x2- √(¹-2)² dx S

A: Solution is here

Q: 4. Evaluate the improper integral dx +e"

A: We know that, ddxex=ex ∫1a2+x2dx=1atan-1xa+c tan-1(1)=π4 tan-1(∞)=π2

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A: Given two functions integrate it by use of formulae

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A: Here first put x=1/t and then the expression inside the square root as y^2. As showing in solution

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Q: Verify that the given differential equation is not exact. (-xy sin(x) + 2y cos(x)) dx + 2x cos(x) dy…

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A: The main objective is to determine the given statement is true or false.

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Q: 3 by taking n= 6 [In x · dx solution of the integral with Simpson's • method 3. (y, + y2 + y4 + y3 +…

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A: We have to check whether the given differential equation is variable separable or not.

Q: 6) Use the successive approximations method to solve the following Volterra integral equation: u (x…

A: The Volterra integral equation is given as u(x)=x22-∫0x(x-t)u(t) dt. We have to solve the equation…

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Q: Consider the ordinary differential equation y"(x) = 4y(x) + 4, y(0)=y'(0) = 1, (d) which is second…

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Q: please explain how to do question 7c

A: The given integral is ∫(1-3x)4dx.

Q: 6) Evaluate the improper integrals ∞ 5 a) 5² (+ + e-") dx s (-x) е 2 7 5 b) ² = 8 dx 4x-8 2

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Q: 4. Calculate the following integrals: 4 3 dx 1 – x2 dz 25z2 + 1

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A: integration

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Q: Step by step

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A: Consider the given integral as Solving the integral

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A: As per the company guideline's we are supposed to answer first question as mentioned by the…

Q: 6. Evaluate the integral e* dx ex + e-x

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Q: 5. Find the particular integral of d²y + dy - x2 + 2x + 4 dx2 dx

A: We use the operator method to find the particular integral of the given differential equation.

Q: Evaluate the integral: e3+ cosh2x33+cosh2xsinh2x dx A (3e)3+ sinh2x 21n3 + C (3e)3+cosh2x B 2ln3…

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Q: 2. VERIFY THATT T7HE FAMILY OF FUNSTIJNS y= 2x²-1 + ce -212 WITH ARBITRARY PARAMETER C !ARE…

A: Compare the given differential equation with the linear differential equation, dydx+Py=Q(x). So,…

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Q: 10. Find the indefinite integral. x – 2 dx x² + 1 -

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Q: II. Evaluate the following integrals: x- 2 1. J- -dx +1 2. J; -dx 8х + 3 3. J- x -4x-9

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