1 dt, for x > 4. In(t) Let F(x) 4

Q: 1. If f(x) cos(t) dt, then f'(x) =

A:

Q: f(x)= /0 x (t3 + 7t2 + 1)dt then f"(

A: The given function is f(x)=∫0xt3+7t2+1dt We have to find f''(x)=?

Q: That the value of the integration g(1) = 0, g(2) = 0 1-e) -dt where g is a function that is…

A:

Q: 2. If f (x) = Sn* et? 0, dt, find f'(x) V2t+sect

A: The solution is given as

Q: If f(æ) = S Tm dt for x > 1, 1 | In t find f'(2).

A: Please refer the attached image for complete solution.

Q: 2. Let f(x, y) = In(x² + y²). Show that fre + fyy = 0. %3D

A:

Q: 1. If f(x, y)=3x' cos y– y sin x– 2.x In y- yln x-4x²y, find ôf ôf d²ƒ _ô²ƒ

A: According to question given thatf(x,y)=3x3 cosy-y2 sinx-2x ln(y)-y ln(x)-4x2y

Q: (a) Suppose that f is differentiable, positive and decreasing on (a, b] and has no critical points.…

A: Given problem:-

Q: Find the minimum and maximum of A over a particular interval [0,6]. A(x) = f(t) dt

A: The solution is given below in the next step: I

Q: fy = g(x) = Ax with In(x)

A:

Q: Q2: Expand f(x,y) = [sin (x y³)]² about (5, 1).

A:

Q: 2z Forz> 0, 순(/"m(P + 1) a) In (t³ + 1) dt For x > 0,

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Q: If f(x) • cos ( z ) t'dt then f'(x) =

A: We can find the answer as below

Q: 4) Determine the function F(x)=[ƒ(1)dt, where ƒ(x)=sgn(x-x²) for xe[-1,2]. -1

A: f(x) is discontinuous but F(x) is continuous

Q: 1 Find f(1) if f(t) dt = x" - x + 2x? – 2, x >1.

A:

Q: 1losamlov ç5x3-1 Differentiate the function g(x) = [°*¯'sin t dt. -3

A: We have to differentiate the given function.

Q: If f* f(t) dt O 6 1 O -4 divergent = 20x √4x²+21 ∞ -4, then f(t)dt is X

A: Given

Q: If 0 < f(x) < g(x) for all x and I f(x)dx diverges, then | g(x)dx diverges. 13 Select one: True…

A: Given the statement

Q: Verify the final value theorem of f(t) =1+et[sint + cost]. O- 1 1

A:

Q: If A(x)= f(t)dt is increasing on [0,10], then f(x) must be increasing on [0,10].

A: Given,A(x)=∫0x f(t) dt

Q: Suppose f(x) = integral -7x   tcos ( 1/1+t^2) dt.  Find f'(x)

A: Given,           fx=∫-7xtcos11+t2dt

Q: (4)Evaluate the definite integral (a)S(1 + sin x )£x 1/2 (b) [ (2t + cost)dt -*/2 x +1

A: As per the Guidelines, I will answer the first 3 Integrals. For the fourth one please repost the…

Q: From the differential equati on by eliminating the arbitrary function ffrom xyz = f (x?+y'+z°).

A:

Q: 5. Given F(x) = cos (t) dt, determine F'(x). 2

A: Using leibniz rule  Explanation is given below....

Q: Use at least two methods to prove that dr/dt=0 when r=√x2 + y2, x=α cos t, and y=α sin t if α is a…

A: First method   r=x2+y2r=(α cost)2+α sint2            since x=α cost and y=α sint  =α2cos2t+α2sin2t…

Q: 1. If F(x)= = ² e²² et² dt, find F(1) and F'(x). ez

A:

Q: If f is differentiable everywhere and f(1)=f(2)=π, then ∫12 f'(t)dt=0   True or false?

A: True because,

Q: 3. Show that for p> 1 and r E (0, 1), the integral below converge: dt tp

A: by limit convergence test we cab prove this, we have to treat the definite integral as summation…

Q: 6. If F(x) is defined by cx²+2x esin(t) dt F(x) = Determine F'(x).

A:

Q: 2. Use the linear approximation of f(x) = x² ln(x + 1) to estimate (1.01)² ln(2.01).

A: We need to estimate value.

Q: 6. let h be the function defines by h(x) = 2e***dt, Find h'(2) %3D

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: 2. Let F(x) = re* %3D VE dt, find F'(x). t2+sint

A: Solution is given below:

Q: Show that f(t) dt ) dx (x - a)f(x) dx.

A: To prove: ∫ab∫xbftdtdx=∫abx-afxdx

Q: 9. Explain why dt r1/I dt 1+t2 1+t? is valid for any x > 0.

A:

Q: Verify Young's Theorem S. f(x, y) = = e(2x+y) In(xy)

A:

Q: d Use the chain rule to calculate f(x) if f(x) In (1+ tº) dt. dx, 0. f(x) =

A: let's use chain rule and find derivative of integral.

Q: if p> 1, (diverges; if p <1. 0o dx Prove that f p- 1 хР 1 Hence evaluate: dx. x6

A:

Q: 1 Find F' (0) given that F(x) : 2 dt. = r In(t+1)

A: Explanation of the answer is as follows

Q: 1 Find F'(0) given that F(x) dt. In(t + 1) 2x

A:

Q: Suppose that i f(t) dt = x2 - 2x + 1. Find f(x).

A: Consider the following equation: ∫1xftdt=x2-2x+1 fx=ddxx2-2x+1=ddxx2-2ddxx+ddx1=2x-2+0=2x-2

Q: | 28. For each x > 0, let G(x) = J each x > 0. e dt. Prove that xG(x) 1 for

A: Given information:

Q: If f(x) > g(x) >0 for X in [1,0) and g(x) dx diverges to +0, then | F(x)dx also diverges to +. True…

A: It is TRUE, that 0&lt;g(x)≤f(x) for x in [1,∞)  and                               ∫1∞g(x)dx diverges…

Q: What value of x > -1 maximizes the integral / t²(3 – t)dt?

A:

Q: (a) If g(x) = (tan t)³ dt, find g'(x). (b) If h(x) = | (tan t) dt, find h'(x).

A:

Q: 13. Prove that the function 2x 1 dt t F(x) is a constant on the interval (0, ∞ )

A: The student has specifically asked for the solution to Question-13. Hence, that question is solved…

Q: Calculate the error of approximation E, for: f(x) = v3x + when x = 8. Ax = dx = 0.1

A:

Q: 4. Express the integral e-t tổ dt in terms of gamma function.

A:

Q: Integrate f(x) = 10 ( 1 – x)9 where 1 ≤ x ≤ 3, Using Boole’s rule

A: The function f(x)=101-x9 where 1≤x≤3. We have to integrate this integral using the Boole's Rule.

Q: Use the Quotient Rule to compute the following: t2 + 1 at t = -2 dt \ t2

A: