21. Let (gn) be a sequence of Riemann integrable functions from [0, 1] into R such that |gn (x)| ≤ 1 for all n, x. Define = f₁" Bu (0) Gn(x) = ܘܢ 8n (1) dt.

Q: Ans. f,=-g(x). f, =g(9) (g continuous for all t) 13) f(x.y.)-x-y+ f.=1. f,--y+. Ans.

A: Given that

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Q: п-(х-4)" 22и. (2n — 1) Ax) =En=0 n:

A: Solution of given problem as follow

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A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

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A: We have to prove the given equation.

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A: Solution is in next step

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A: given function is                                                        g(x)=lnx2+2 log is defined…

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A: This question from complex analysis' s topic entire function.

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Q: (c) Evaluate the Riemann integral tion Pn ={1," i=1 n+1 n+2 n' n 1 dx by using definition with the…

A: Hint: Use the formula ∫abf(x)dx=limn→∞b-an∑i=nfa,ib-an to obtain the required solution.

Q: Find all continuous functions f(x) defined on x > 0 with f(x) = 0 such that f(x)³ = f(t) dt a

A: