15x? – x3 – 43x – 19 - - The integral | can be rewritten as (x + 1)²(x² – 6x + 13) Ах + B + lax² + bx + c ' (mx + n)t ' rx +s] + Ax + B ? ax² + bx + c What is the value of () žAretan () (a) Arctan + In|x² – 6x + 13| 1 (6) Arctan () (2)- In|x² – 6x + 13| (d) In]x² – 6x + 13| 2

Q: True or false: The coefficient of x4y7 in (3x – 4y)11 is (2)347. True False

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Q: 2) i)' If 1 [ f(x) f(z)dz = 6 ln(|z|) sin(z) - 3e³ +2². then find f(x). ii) { Evaluate se - r² + 16x…

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Q: Let f(x) = x' + x + 4 and g(x) = 3x + 2 in Zs[x]. Then, upon dividing f(x) by g(x), we get O The…

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Q: 2 - Jx - 3 – v 4) lím エ→7 x² – 49

A: Given: ⇒limx→72-x-3x2-49

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Q: Find of z = In(xy)

A: Given,Z=ln(xy)differentiating both the sides partially w.r.t. x∂z∂x=1xyy1∂z∂x=1xagain differntiating…

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A: We have,

Q: 5. If f(x) = (x + 1)(x² – 2)³ then f'(x) А. бx (x? — 2)? В. бx (х + 1)(х2 - 2)2 C. 3(x + 1)(x² – 2)²…

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Q: Compute 1 (x² + 1)(x² + 4) Lo -dx.

A: Given ∫-∞∞1( x2 + 1 )( x2 + 4 )dx

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Q: 4. Use residue theory to calculate the following integrals: 3x (a) P.V. · Loo (2²+49) (2² - 2x + 5)²…

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Q: f (x) = { ₂x² (²/²) ² 2=1 Coso f(z) dz= ? 23

A: We have to calculate

Q: x-1 6)/2=0²+5² x² - 4x + 5 (1) (k) / zeaz² +b dz dx (i) (2x² – 5) sin x da (1) √²×² = √16x² dx

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Q: A Moving to another question will save this response. Question 10 "The product of a negative real…

A: Let us assume a negative real number x. So, x<0 And a positive real number y. So, y>0

Q: 29 x) = [42] (4), x(t) x(o)= [?]

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Q: None

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A: Perform integration

Q: 5. If f (x) = (x + 1)(x2 – 2)3 then f'(x) da व्य 2= 2x (1) 2 A. & 6xx2 – 2)2 انی + (22 21 2 = Alv 3x…

A: Given query is to find the derivative of the function.

Q: Evaluate cos (x (x2 – 2x + 1) (2x - 2) sin (x 2 - 2x +1) (2 – 2x) sin (x? – 2x + 1) O (2x 2) cos (x-…

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Q: If f(x) = 3x² +5+ (4x – 3)* | , then f '(x) is - + 5+ (4x – 3)*]" [4(4x – 3)'] B. [3* +5+ (4r -…

A: Given:            fx=3x2+5+4x-349 Solve: f'x           ddxfx=ddx3x2+5+4x-349

Q: Integrate the function. 2+81 dx 4x2 V In Vx2 + 81 + x x2+81 4x + C In Vx2 • 81 + x] Vx2 + 81 + C 4x…

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Q: 10. y"of y = arc tan

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Q: -Vx2+5 -x f(x): f'(0)? -x+2

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Q: Demonstrate that x' + 3x – 8 is irreducible over Q. -

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Q: Consider the following bijections on R: f (x) = - g(x) = x – 7 Then (f o g)¯'(x) =

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Q: Show that f(x)= x14 – 6x + 75 is irreducible over Q.

A: Eisenstein's Irreducibility Criterion: Let us consider polynomial…

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Q: 07/ Express in the form x + jy (1) e'(x/2-j) (2) In(3 – j) (3) sin(r/4+ j2) -

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Q: d dx [erf (x)] = = = = = = = = x

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Q: F(x+y=z₁ x² + y²) where 2= z(x,y) Prove that x2² - y2/²/² = x-y dy if it was

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Q: eſt 2 du given that * f(z) dx = 12 f(4Tan¹(u)) 1+u² Evaluate

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