If the integrand contains: a² x² √a² + x² √x²-a² S 1+x² X Apply the substitution: Subject to the restriction: ㅠ 70 x = asin 0 ·≤0≤ -dx x = a tan 0 x = a sec 0 - 0≤0< or π ≤0< NENE In problem do the following. (a) Choose the appropriate substitution for r in terms of using the above table. (b) Determine dx in terms of de. (c) Draw and label a triangle with acute angle 0. (d) Apply your substitutions and evaluate the resulting trigonometric integral. 75 NI 3π 2

Q: 2.2 y = (x³ − 1)(x² − 2x + 1) using the product rule. √1 + tan² x 2.3 y= sec² x

A: solution of the given question is like aS:

Q: On which interval f(x) = -2³ + 3x² - 4 is concave upward? O a. (-1,1) O b. 1 Od. >1

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Q: 2. Show that by using MEAN VALUE THEOREM . u-v < tan-1 v - tanu < if 1+v2 u-v 1+p2 0< u<v and ded…

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Q: Prove the left side equal to right side in the following : 1 L-1 (s² + a²)² [sin at + at cos at] 2a

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Q: 10. Evaluate 1 Hint: Substitute x = 3 sin 0, where - T/2 ≤0 ≤ π/2. √9-x² x² dx.

A: The given integral is

Q: Given α+ßi = √¹x³¹_x=¹¹dx, Inx 0 α, ß ER, find the analytic expressions for a and

A: Recall the following formula  ln(a+bi)= ln √(a²+b²) + i tan-1(b/a) .

Q: OYm Sin (x-1/6)

A: Laplace transferom

Q: C. lim x→1+ 5x 4 31 -1-x 3x]

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Q: Show that the extreme values of f(x) = a sin x + bcos x tVa2 + b2. are

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Q: If I, = [x^tan"x dx, Prove that (n+1), + (n-1),,2 ° п 1 %3D 2 n

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Q: In the expression V x2 - a2/x (a > 0), let x = a sec 0, where 0 < 0 < ", and simplify the result. 2

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

A: To find the value of given integral.

Q: a. as x→. tanx- b. as x→ tan x→ E. as x→0", cotx→ d. as x→0*, cotx→

A: Given query is to find indicated limits.

Q: 16. Evaluate ft tan5 x sec¹ xdx.

A: We know that Sec²x = 1+ tan²x ∫ xn dx=( xn+1)/(n+1)

Q: 4.x sin (In(x² + 1)) dx Evaluate I x² +1

A: In this problem, simple substitution formula is used to solve the integration

Q: 5x C. lim 1-52, x→1+ -1-x 3x] D. Compute the derivtive of sin³ (x + 1)

A: Given: sin3x+1. We have to compute the derivative of the given function.

Q: Select one: a. O b. O c. dy dx e. dy dx dy dx 3 ind the derivative for the function f(x) = 3x³ + 2x²…

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Q: | a 0.24) tan(sin-1x) = b) c) =x d) 1 - x2 3

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Q: Find For the Following FunctIon XP 8x 2X8 22

A: we have to find dydx for the given equation.1. y=6z2+18x4-12x8differentiate w.r.t…

Q: Find the definition set of the following function F(x)= √ln(sinπx)

A:

Q: 4. lim -0 A. -2/2 B. 1 C. 2 h D. Does not exist sin

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Q: If = {(x, y) | 16 ≤ x² + y² ≤ 49} then: 11.000 = [ co cos (x² + y²) dA

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Q: If 4x<9(x) <2x-2x°+4 for alle x, evaluate lam g(x). Explain!

A: on solving this we get

Q: 9. The value of f 1-x 1+x A. sin ¹¹x + √1 + B. sin¹x - √1 + dx x² + C x² + C C. sin ¹¹x + √1- x² + C…

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Q: Assuming that |a| < 1, find the value of cosh ax dr. cosh x

A: It is given that,a&lt;1, we have to find the value of              ∫0∞coshaxcoshxdx

Q: Which of the following is NOT a root of the function f (x) = x³ + 4x² – 7x – 10 ? а. -5 b. -1 с. 2…

A: We will solve the following.

Q: For how many n = {1, 2, ..., 695} is n a multiple of one a multiple of one or more of 8, 3, or 7?

A: In the range of [1, 695] the number of numbers divisible by an integer k =  represents the floor…

Q: . Evaluate f dx using trigonometric substitution. x/4+x2

A: Given integral is ∫1x4+x2dx. We have to evaluate ∫1x4+x2dx by using trigonometric substitution.…

Q: 11. If f(x)=-4¹, g(x)=√√x and h(x) = f(g(x)), determine h(-4)? ✓✓✓ Si7s100

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Q: (c) (z-a)", where a EC and n = N*

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Q: Prove that there is a real number r so that sin(r) = x – 1.

A: We have, sinx=x-1 Let us consider a function fx such that, fx=x-1-sinx

Q: Variable Separable and Homogenoue D.E -x + x? - x - Sin? dy (3x-4x? - っ) cosydx

A: We will find out the required general solution.