75. dx xex +x2 Hint for Exercise 75: Show that for 0

Q: 21. f(x) = In (5-3x²)' J6x² + 21. f(x)= In| (5-3x V6x? + 8х —12 1+5t 22. f(t)= In(t)

A: As per our guidelines, I can answer maximum one question. Since you have asked for multiple…

Q: 9. 3 If f is continuous and f (u) du = 3, then / (4xf(x²) + f(3x)) dx is equal to

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Q: Use Simpson's 1/3 rule with (n₁ = 2), and Simpson -2 1 estimate tan-¹(2). [Hint: tan-¹ (2) = ₁²; dxl

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

Q: 8. If g(x) = e*In(x+1), compute g'(x). What techniques are you using in this case?

A: Solution is given below..

Q: Starting from the function f(x) = e¬alx| -L 1 where

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Q: Let L(x) = ln(x² + 4x + 14) in the interval [-10,10]. What are the absolute extrema of L(x) in this…

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Q: B-Test and proof the function So (x In x)² dx

A: ∫01(x logx)2dx=∫01x2 (log x)2dx use integral by parts, ∫udv=uv-∫vdu u=(logx)2…

Q: 5.Use trapezoidal rule to evaluate f,(x² + 1)7 dx by taking n = 4.

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Q: 00 Suppose 0 < f(x)< g(x) and both are continuous on [a, o0). If | 9(x) dæ diverges, 00 | f(x) dx.…

A: We have to apply the comparison test

Q: 73. Use the definition of |x\, the chain rule, and the fact that (In x) = - for x > 0 to prove that…

A: Given, dlnxdx=1x, x&gt;0 from definition of x x=x ,     x&gt;0 -x ,  x&lt;0  if x&gt;0 then d…

Q: find the critical points:

A: Given function is  and the interval isThe critical points of a function are obtained when the first…

Q: dx Evaluate x²/4 x²

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Q: 64 8. f(x) dx = 14, find xf(x?) dx. If f is continuous and %3D

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Q: 1. Suppose that f f(x) dx = In x + Vx+1+C. Can f be a rational function? Explain.

A: To explain f is a rational function or not.

Q: Q1- Prove that coth x = 2 csch 2x + tanh x

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Q: 81 xf(x²) dx. 6. If f is continuous and f(x) dx = 4, find

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Q: Find number k so that f(z) = 10x + 8 x -1 is continuous on (-0,0). O -2 4 -4

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Q: 13)Evaluate 13 Evaluate the integral s 5x--Xt27 dx x (x²+G) -1 3 tan 3+C b)(-ト2フx)、1mlxCx?ss)1c Sx3…

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Q: 4 – x² p(x) = x – 1 at x = 0 Select the correct response: Continuous O Discontinuous

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Q: 4х+2 Evaluate the integral -dx. (Hint: Use Algebra to help further rewrite your (x² +1)(x=1)…

A: First break by using partial fraction then integrate

Q: 36 6. If f is continuous and f(x) dx = 2, find xf(x²) dx. %3D

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Q: 81 9 | If f is continuous and f(x) dx = 8, find xf(x²) dx.

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Q: 3. Find the value of k that will make f continuous on (-o, 00). NORS x + 19 if x 1

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Q: 25.) Use R3 to find a bound on error when P3 = x – x³ is used to approx f(x) = sin x on the interval…

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Q: Find the linear approximation of f(x) = ln x at x = 1 and use it to estimate ln(1.34). L(x) =…

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Q: Given f(x) = ln(ln æ³) Diberi f(x) = ln(In x³) %3D 3 Show that Tunjukkan 1 f' (x) = xlnx %3D

A: Given that the function is fx=lnlnx3 and it is required to show that f'x=1xlnx.

Q: '12 9. If f is continuous and f(x) dx = 20, find f(2x) dx.

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Q: 3. Use the IVT to prove that y = x ln x has at least one zero on 2'2

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Q: Find the Padé approximation R1.2(x) for f(x) =(ln(1 + 2x))/x

A: Pade approximation R1,2(x) for f(x)=ln(1+2x)x…

Q: 18 9. If f is continuous and f(x) dx = 14, find f(2x) dx. !!

A: Solution is in step 2

Q: 64 8 If f is continuous and f(x) dx = 18, find xf(x²) dx.

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Q: Suppose F(x) is between 4 and 8 for x in [3,12]. What is the largest that · 12 L" F (x) dx can be?

A: given that function F(x) between 4 and 8 for x in [3, 12] we have to find the largest value of…

Q: Use the Intermediate Value Theorem to show that 2x – 3 - if – 3 0 has a zero on the interval [-2,…

A: First compute the function if they are continuous then apply the intermediate value theorem.

Q: 16 8. If f is continuous and f(x) dx = 18, find f(2x) dx. %3D

A: Given integral, ∫016f(x) dx =18

Q: a In(1-x) X lim a-1 So dx

A: We can solve this using series then integral

Q: 5. Evaluate f(0) when f(x) = 2 tanh x. А. О В. 1 e²+1 2e D. -1 E. None of the above 2e

A: To find f(0)

Q: Q7: Use the MEAN-VALUE-THEOREM to show that x-1<e* < 2x+1, 0<x<ln 2.

A: Since you have asked multiple question, we will solve the first question for you as per our…

Q: 9 3 Iff is continuous and f(x) dx 8, find f(x2)dx

A: Given as

Q: Determine the value of the arbitrary constant of the antriderivative of F(x) = x-In(x) given the…

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Q: '18 6. If f is continuous and f(x) dx = 16, find f(2x) dx. %3D

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Q: 13) Evaluate div F where [[x, 4₁27 = <-xy, x², 2³ e xy² ) 32 *** - ) 2xyz e ) 242 243 - ۶۶ ) *+3x 3…

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Q: Q4: To evaluate the integral fcos' xdx using the reduction formula of cosinc function

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Q: 19. f(x. y) In(1 - x - y) %3D

A: As per our company guidelines we are supposed to answer only the first question that you have…

Q: Using Simpson's (1/3) rule with h = 1, evaluate the integral: 7 = f x² logx dx 3

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Q: 9) Given the function f(x)=-3x+2 , find the net signed x-axis over the interval [-5,3]. area between…

A: Given function is  f(x)=-3x+2 We have to find the area of the region bounded by the given function…

Q: 22 + a For what value or values of k is f(x) = ka + 3k - 5 continuous at a = 5?

A: NOTE: Refresh your page if you can't see any equations. . to find the undefined point, take…

Q: The Gamma Function is defined by [(x) = fx-¹e-tdt, positive integer then prove that for na T(n) = (n…

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Q: Find 뿐 for the ry – ln(xy) = 1 0 뽈 ㅇ-볼 O 래 ㅇ-픕 following:

A: The given expression is xy-lnxy=1 we have to find dydx use the derivative formulas, ddxx=1…

Q: k. f(x)=cosh(x³)+ sinh(10x) 1. f(x)=In(x* +1)

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.