Use mathematical induction to prove that the n-th derivative of (x +n + 1)e" is given by d" (x+1)e = (x +n+1)e* dxn for every positive integer n = 1, 2, 3, ...

Q: d {In(1+x + x²)} Find dx

A:

Q: Let n be a positive integer. Prove that 1:(1)+2 (;) . (:)- n n + 2 . +n . n2"-1. ... n

A: We have to prove the identity: 1·n1+2·n2+....+n·nn=n2n-1

Q: 1 -dx 1-x²

A: This particular question will be solved by Partial fraction: Solving by Partial fraction:   Partial…

Q: Q1 Let X1, .. ,X, f(r;0) = ()0 (1 – 0)2-, x = 0, 1,2. (i) Show that f(x:0) is a member of the…

A: It is an important part of statistics . It is widely used .

Q: What would be the E[x]?

A: The probability distribution table is: X P(X) 0 0.2121 1 0.5091 2 0.2545 3 0.0242

Q: Apply the special case of the general power rule d/dx [h(x)]2 = 2h(x)h(x) and the identity fg =…

A: dhx2dx=2hx·h'x  &  fg=14f+g2-f-g2

Q: 2(д-1)(1-и) 9-1)(2-и)и+1 log((q-1)(2—и)и+1) u? > 0 for u > Show that u E - 0, 1], q > 3

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Q: Show that

A:

Q: In x

A: A detailed solution is given below

Q: / In (x? – x + 2) dx — х+2) dx

A:

Q: 7. [(In x)² dx

A: Given- ∫(lnx)2dx We have to evaluate the given integral. Here we will use the following formulas- ∫u…

Q: [UCONN 1071Q] Let f(x) = (e+ln(x2)). Which of the following represents the first derivative f'(r)?…

A: .

Q: (a) Use the definition of L{·} to determine L{te=#}

A:

Q: In(x) x² 1 5,00 dx

A:

Q: Show that S, (In x)? dx = e – 2

A: The detailed solution is as follows below:

Q: Q3) Find the derivative of y for all functions the following (1) y = 0 (sin (In 0) + cos (In 0)),…

A: Since you have posted multiple questions, we have solved the first question for you, if any specific…

Q: Let f(x) = In[x4(x +9)6(x2 + 4)3]. Find f'(x). f'(x) =

A: Given,

Q: -5x Integrate x² e

A: We have to find

Q: What is the choice of substitution for S e*(1+e*)(1/2) dx?

A: The given integral,        ∫ex1+ex12 dx

Q: Show that esin(x²)dx < e– 1.

A:

Q: ! Given f (x) = {x + 2 if x > 10 lx – 7 if x < 10' D. ! determine each of the following. a.) f(14)…

A:

Q: Let f be a decreasing function such that f(0) = 4 and f'(0) = h'(0)=; 3 Oh'(4) = 3 Oh'(4) = 0…

A:

Q: dn Find a general formula for den by identifying a pattern in parts (a) through (c). To begin,…

A: We use the given details and differentiation formula to find answer

Q: 3. Prove that -2/3 dx diverges by showing that х lim -2/3 dx = 00 х

A: Given limit is Since one of the limit of integration is infinite , so we evaluate the limit as…