2 In z +1 gian( dz -de cos? (Sz) 2 in |(ln z)* + In a| +C la (ma) + laz +C In 3 In 3 glane) (tin a) + laa| +C In (in ) + In a+C 5In 3 5 In 3 (sin z+ cos z) sin z zp-

Q: If e 2i =sin2 then z = = sin2 = 2 - 2 In(sin2) + 4kT 2i In(sin2) + 4k TI - 2i In(sin2) – 4k T | - 2i…

A: Given that, e-z2i=sin2e-z2i=sin 2e-z2i×ii=sin 2eiz2=sin 2

Q: Determine and of y = (5r - })(cos z + In z

A: Consider y=5x2+32sin x+ln x. Calculate: dydx=y'. Apply product rule of derivative:…

Q: /2 1lf If In = " x cos" x dx, when n> 1, show that п(п — 1)1-2 — 1 In 12 [Hint: cos" x = cos²…

A: Given: In=∫0π2x cosnx dx

Q: If e /2i =sin2 then z = 2i In(sin2) + 4k TI - 2i In(sin2) – 4k TI - - 2i In(sin2) + 4k TI - -2…

A: Use the principal log to find z.

Q: The area of the region inside r = 2+ cos 0 and outside r= 4- 3 cos 0 is given by T(2 + cos 0)? – (4…

A: We will find out the required area of the bounded region.

Q: 2 2 In sin 2x - 2 In (1 + cos 2x) = In ( )- %3D

A: Given that: The identity is 2 ln sin 2x - 2 ln ( 1 + cos 2x) = 2 ln tan x .

Q: s nip

A: Given z=tanα2

Q: The area of the region inside r= 2+ cos 0 and outside r = 4 – 3 cos 0 is given by I [(2+ cos 0)² –…

A: First find the angle of intersection of given two polar curves,…

Q: Find * sin (In æ) cos²(ln x) dx. cos (In x) , cos³ (In x) 3 cos (In x), cos³ (In x) 5 3 5 Option 3…

A:

Q: uestion 4. Evaluate the value of the integral 2. COS cos (30) dr. 5- 4 cos(20) 2x

A:

Q: iii) E(-1)"+1 cos²(÷) COS n=1

A:

Q: 14. If we let u = 4x – 1 for the (2tan(4x-1)-3 cos“(4x=1) dx. what will be the cos(4x-1) resulting…

A:

Q: Z1 "(cos(0, r2 Z2 02) + isin(0, – 02)](z2 # 0). 85. Let z1 = ri(cos 0, + isin 01) and z2 = r2(cos 0,…

A: Given z1=r1cosθ1+isinθ1 and z2=r2cosθ2+isinθ2.

Q: the integral F(®) = cos*(0) + sin(20) – 3 cos(0) COS do sin (0) +3 cos2(0) + 2 sin(0) – 3

A:

Q: sin 40 Evaluate the integral : do 3 + cos 40 using the substitution u = 3+ cos 40. sin 40 do = 3+…

A: Solve by substitution method

Q: In Problems 28–44, establish each identity.

A: Trigonometric identities are given, we have to prove the identities. Note: we are answered with 3…

Q: sin In 4+tan cos-dx 4+tan-Icos- 2 1+ cos 2sin In4 4+ tan cos- dx A 1+cos 4+ tan-cos- 2sin-In 4+…

A: To rewrite the given integral so that the integral can be evaluated easily. ∫sin…

Q: (25) Let k and n be positive integers. Prove each of the following: sin (n2) dz = | cos (nx) dz = 7.…

A:

Q: sin(k/n). 1. cos(1) + 5/3. 2. cos(1) + 1/3. 3. cos(1) + 2/3 4. cos(1) – 1/3. O 1 O 2 O 4

A:

Q: Show that (cos x + isin x)n= cos nx + isin nx whenever n is a positive integer. (Here i is the…

A: We can prove given result by mathematical induction: Define a statement p(n).  Show that p(n) is…

Q: 5s Evaluate J|2 sin s Jdtds 0 0 I[ 2 OA. 3 (1 – sin(125)) O B. 3 (1+ sin(125)) (1+ cos(125)) 3 OC. 2…

A:

Q: Theory and Examples

A: limx→02x23-3cos x

Q: The area of the region inside r = 2+ cos 0 and outside r = 4 – 3 cos 0 is given by | ((2+ cos 0)² –…

A:

Q: 2. Prove the identity: cos (x-y)– cos (x+y) = 2 sinx siny

A:

Q: 3. Find the area of the cardioids region shared by ri 7(1- sin 0) and r2 = 7(1+sin 0). A 7(1 Sint) B…

A: The given curves are:  r1=71-sinθr2=71+sinθ To find the intersecting points,…

Q: Show that n. In(cos 2k+2) = - In (2"+ sin 2n+2). k=0 Hint: sin 20 = 2 sin 0 cos 0.

A:

Q: 5- Show that a) sinh (z + ni) = – sinh z c) tanh(z + i) = tanh z

A: The complex function is of the form z=x+iywhere x is the real part and y is the imaginary part. A…

Q: v3 V3 sin 30 = , cos 30° = , tan 30° = COS %3D O sin 30° = , cos 30° = . , tan 30° = %3D %3D V3, cos…

A:

Q: 1+cos 8 ()- Chow that cos and sin 1-Cos e %3D using de Moivre's Theorem. %3D

A:

Q: 3. tane de ) 71+ n cose)/3

A:

Q: Integr S√₂-cod -1 ation fuctions. 1-cos²x) dx Not of the above B 1 O

A:

Q: Which of the following is true? A Jsin(3x)cos(4x)dx= [ Sin(7x) – sin(x) Jcos(3x)cos(4x)dx= [cos(x) +…

A:

Q: 7.2 (A) 1. S sin x dx cos x + C because(-cos x +C) = %3D dx 2. S cos x dx = because sin x 3. S tan x…

A: We need to solve the given integral. This can be done as follows:

Q: • 8 If In = 7/2 e*cos" x dx, where n 2 2, prove that: %3D (a) In = 1-n[" /2 e-* sin x. cos"-1 x dx…

A:

Q: a Show that (sin(m + n)x + sin(m – n)x) b Hence, without using the product rule, differentiate sin…

A: We will find out the required values.

Q: 11. One leaf of r = cos 30 12. One leaf of r = sin 40 13. Bounded by r = 2 cose %3D 14. Bounded by r…

A:

Q: Prove that (a) Pn (cos 0) = 2F1(-n,n+ 1; 1; sin²0/2).

A:

Q: 1 Evaluate | (x²sin y- xcos(y))dy O 2sin y-3cosy + C 6 -3sin y+2cosy +C 6 x2(1- cos(1)) – xsin(1) +…

A:

Q: 7π/4 Evaluate the integral (sin x+ cos x) dx using the fundamental theorem of calculus. Discuss 2+…

A:

Q: Z {sin(3t + 2)} O (5 sin (3) + 2cos (3))/ (s^2 + 4) O (s sin (3) + 2cos (3))/ (s^2 + 4) (s sin (2) +…

A:

Q: sin(k/n), lim is 1. cos(1) +5/3. 2. cos(1) + 1/3. 3. cos(1) +2/3 4. cos(1) - 1/3.

A:

Q: [6z" cos(2) dz =- O 6 r* sin(x) – 18 z² cos(x) – 36 z sin(z) – 36 cos(z) + C O 6z° sin(z) + 18 z²…

A: Explanation of the solution is given below...

Q: derrivative of: Cos(xy)+secy=5

A: Consider the given expression as cosxy+secy=5

Q: lim (1 - cos(3x))sin(7x) of (c) 이이 O 0° O 100 O o. c0 O co - c0 O not indeterminate 8 %

A:

Q: (6) Given that In= tan" x dx, show that, for n 2 2. %3D In- In-2 = %3D n- 1

A: I think given question is slightly wrong. According to me the required proof should be…

Q: find 7. (x, y)→(0,ln 2) In |1 + x² y²| lim et-y 8. lim (x, y)→(1,1) 9. e' sin x 10. (х, у) —-(1/27,…

A:

Q: f5cos(1-5x) dr = O - 5sin (1 – 5x) + C O -sin (1 – 5x)+ C 一 sin (1 5x) + C 5 - O sin (1 – 5x) + C…

A:

Q: 1+cos 2x dx sin 2z O cot 2x - csc 2x + c O cot 2z +In |sin 2z|+ c O none of these O In (csc 2x + cot…

A:

Q: 2. Prove that cos30 = (4cos'e - 3cos0) and sin30 = (3sine – 4sin e) using De Moivre's Theorem. %3D…

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.