91. Let I, = | x" cos(x³) dx and J, = | x" sin(x²)dx. (a) Find a reduction formula that expresses I, in terms of Jp–2. Hint: Write x" cos(x²) as x"-'(x cos(x²)). (b) (c) Evaluate 13. Use the result of (a) to show that I, can be evaluated explicitly if n is odd.

Q: 2. a) Find a formula for f-1 for the function y =f(x) = (x – 2)2 , x< 2 b) Evaluate the expression…

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Q: Differentiate the following functions using the quotient rule. a) You are given that f (x) a df dx…

A: The Power Rule:The Chain Rule:The Quotient Rule:

Q: Use the product rule to differentiate the function h, where a and b are real numbers. h(x) = e¯ax…

A: According to product rule, if f(x)=u(x)*v*x) then f’(x)=u(x)*v’(x)+v(x)*u(x).

Q: 2. If f(g(x)) = sin x + 3sinx - 1, determine possible expressions for f(x) and g(x), where g(x) # x.

A: (1st method/way): Let f(x)=x²+3x-1 and g(x)=sinx Then we can find f(g(x)) as follows…

Q: Let f(x) = sin2x and g(x) = cos2x. Determine whether h(x) is in span(j(x), g(x) ) h(x) = cos 2x

A: Given  f(x) = sin2x, g(x) = cos2x and h(x) = cos2x Then, we have to determine h(x) is in span(f(x),…

Q: 5. Evaluate: 34 l vo dx dx. √9-16x2 0

A: To find the value of given integral.

Q: Which function is obtain by shifting the graph of f(t) = sin 3t a distance of 3Tt/2 units to the…

A: Given f(t) = sin(3t)  A shift of a units to the right of t axis is given by  f(t-a)  Thus a shift of…

Q: Evaluate L{t°e! e3(1 – t) 3e-t sin (4t - 2) 2) + t2 $ (3t)} cos

A: Explanation of the answer is as follows

Q: Differentiate the following functions. Simplify and factor your answer where possible. a) f(x) = e2x…

A:

Q: ifferentate: f(x) = COSX 1+sin x Apply the quotient ru

A: Topic:- application of derivatives

Q: ution t= tan 2 tan). os(x)

A: We have to choose the correct answer from the given option

Q: Q1) Evaluate: (a) fe²_1_dx Inx sin ¹x 1-x² dx (b) f 1- cos dx (e) Jỗ 2sinx cosx dx (c) | Inx dx

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Q: Find all xE[0,2pi] that satisfy sin 2x + cos x = 0 x =

A: Topic- Trigonometric Equations We are given, sin(2x)+cos(x)=0 and x∈[0,2π] We know that sin(2x)=2sin…

Q: 1. Differentiate. a) f(x) = x43x b) g(x) = x² — e√x -

A: We need to perform differentiation.

Q: 4. Prove that cos(sin¬1 x) = /1 – x².

A: cos(sin-1x)=1-x2⇒sin-1x=cos-1(1-x2)Let y=sin-1x                               (1) ⇒siny=x…

Q: 1. Differentiate. Do not simplify. а. f(0) = 3cos(0*)sin(0³ – 2) b. f(x) = 4x + 25-x² + 2x-1

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Q: a) f(x) = 1+ Sinx b) f(x) = x² + 2x- 1 c) f(x) = 2* + 2 for all x eR for all x eR for all x eR

A: Given function are: [1] f(x) = 1 + Sinx   for all x ∈R [2] f(x) = x2 + 2x - 1  for all x ∈R [3] f(x)…

Q: dx Evaluate: /9 – x² A @ sin -1 +C .3. 1 3 sin +C 3. B

A:

Q: 3. Determine the exact value of f"(0) for f(x) = (cos5x + 5x²)*.

A: Given

Q: 2. Use the identity of inverse hyperbolic function to solve cosh-(2 – a2) = ln 2, for |æ| < 1.

A:

Q: Differentiate the functions. a) f(x) = 2 – 5x + b) f(x) = e* + cosx c) f(x) = d) f(x) = (5 – x²)· e*

A: Suppose f and g are differentiable function .Then note the following results : The power rule of…

Q: 6. Find : (a) y = (x³ +5) tan 7x, %3D

A: