7 a Show that for all positive integers n: 27 2n sin nx dx = 0 ii cos nx dx = 0

Q: 20 a Show that sin (A + B) + sin (A – B) = 2 sin A cos B. b Hence find: i 2 sin 3x cos 2x dx ii sin…

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Q: Z and dy 없 of y%3 (5x2 - ) (cos ar d²y Determine (cos x + In x).

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Q: Step by step

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Q: The f sin 2x cos 2x dx is 1+3 cos 4x 1 sin²4x + C 24 1 In|1 + 3 cos 4x| + C 12 {A} {B} {C} - In|1 +…

A: To find the value of the given integral ∫sin2xcos2x1+3cos4xdx using substitution

Q: B\ Suppose that = 4 cos? t , x = v3 sin(2t) and 0 < t < n. S d²y *= - tan(2t) and = -sec (2t). 1 dt2…

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Q: The approximation of 1 = f*cos (x5)dx using composite Simpson's rule with n = 3 is: 0.5412 O None of…

A: Option A is correct

Q: The integral / 2/ (*) cos(2*) dx takes the form k cosu du when u=2* and f (x) =x – 5. what is the…

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Q: Evaluate the integral. 1/2 | [2 cos ti- 3 sin 7t j+ sin? 2t k] dt

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Q: Can i get help step by step with this problem?

A: Recall the following facts.

Q: Evaluate the integral:: 1. § 4+0²³²- dz e² · cos³x + 8 sin x cos²x + 6 sin²x dx ; cos x ‡ 0 2.5.…

A: We have to find value of given Integral.

Q: 2. : (a) For n ≥ 1, show that 1+z+2²+...+ 2¹: (b) Hence, show that 1+ cos 0 + cos 20 + 12+1 2 #1 7…

A: Solution:

Q: Evaluate the following integral. 3 4 4 cos '3x xp. sin 3x 1 18 4 4 cos '3x dx = V1- sin 3x 18 (Type…

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Q: To show that rπ/2 sin" x cos" x dx: (m - 1) (m- 3) ... × (n – 1) (n - 3)... (m + n) (m + n - 2) (m +…

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Q: "ind no E N such that for all n > no sin(n + n² cos(n²)) n3 – 5n4 – 20 -

A: Find n0

Q: Evaluate the definite integral: 7/4 1+5 cos2 (t) dt cos? (t)

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Q: 9. Let r = 3 sin tî + 3 cos tj + 4tk. Calculate N

A: Given r(t) To find N(t)------ unit tangent vector

Q: /2 1. (a) Use the Wallis cosine formula to evaluate cos' x dx. (b) Evaluate | 1 - dx by using…

A: Wallis's formula state that 1. if  n is odd and n≥3,then ∫0π2cosnxdx=23.45.67 . . . n-1n Therefore,…

Q: 1. a. Let In = exp (nx) sin (exp (x)) dx. By making an appropriate substitution, or otherwise,…

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Q: 4. a. Show that for 0 Va² + sin² a. b. Generalize the result in part (a).

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Q: Given that a, b and n are positive real numbers. Write down ONLY the answer of the indefinite…

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Q: Show that COS x dx a² – x² sin a 2а where a is some positive number.

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Q: 2. Evaluate the definite integral f²n 15 sin*(3x) cos(3x) dx

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Q: Prove that: cos(nx) - cos(na) cos(x) – cos(a) sin(na) sin(a) dx = n neN, a ER - {0}

A: To solve this question take different values of n and calculate LHS and RHS separately.     if…

Q: nd the definite intergral upper 1 lower 1/2 sin-1(x)/ square toot 1-x2 dx

A: Integrate ∫121sin-1x1-x2dx                          ......................(1) Put sin-1x=t…

Q: 7/2 79. Prove that 0< sin x dx < Ja/4

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Q: 14. The Integral:√4-3x² a. cos²0.de dx, can be transformed into : b. 2f cose.de c. 2 f tano.de d. 4…

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Q: The integral / 2' )cos(2*) dx takes the form k cosu du when u=2* and f (x) =x- 5. What is the value…

A: we use the formula ddx(ax)=ax lna

Q: D. 1. If (x + y)² = x² + y² 2. If y = sin³ () cos³ () dy - dx X prove that y' = sin² (x) cos(x) 8…

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Q: Q3/ Determine: (a) sin(t) .8(2t – 1) (b) (cos nt). 8(t- 1)dt 8(t –

A: As per our guidelines we supposed to solve the very first question if the multiple questions are…

Q: Determine the value of the serie 4∑ ((-1)^n)/(n^2) *cos(nx) at the point x=1

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Q: 5. (a) Recall that a' = a' · In a for 3' dt any a > 0. Evaluate dt . 1 + 32i (b) Evaluate (cos…

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Q: 8. 6x° cos (x³) dx

A: ∫ x8cosx3dx

Q: 7) Evaluate the indefinite integral / cos cos 2t sin 2t dt.

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Q: 5. Evaluate 1. S, sin(x2) dx

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Q: 1. Evaluate the integral 7/2 cos xesinx *dx

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Q: to the I. sinx(e+ 1)x' - cosx = 1, y(=) = 0 %3D %3D (1 + e) (1 + cos x) = 0 (1 + e) (1 + sin x) = 2…

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Q: 1. Express as a sum or difference a.) sin 7t sin t b.) cos 6u cos (-4u) c.) cos 2t sin 6t d.) 2…

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Q: 3. Show f cos2n Ode 277 sin-n Ꮎ ᏧᎾ = 27 22n 11.1 n = 1, 2, 3,...

A: Using Cauchy integral formula.

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…

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Q: F.(3e-3 sin(2x')) is equal to 2 3. n 4+@² 3 cos 8 4 3. 7(1+o² 4 cos 4 4 1 3, 1 +4cos 4. 4 2 1+@ 4…

A: Use the following formulas to calculate the Fourier cosine transform. Fee-ax=2πaa2+ω2…

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: 4) Differentiate each of the following: a) y = - cos²x c) f(0) = sin(20 — π) b) y = sin(20) - 2…

A: Differentiate the following  a) y=-cos2x b) y=sin2θ-2cos2θ c) fθ=-π2sin2θ-π

Q: Solve the eaation on the interval. 2sin Đ =3(cos o t D

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Q: (b) Prove that 2 -3/2 Js12 (x)dx = sin xy + C. 'COS X x2 x3

A: We have to prove ∫x-32J52(x)dx=2πcosxx2-sinxx3+C

Q: The f sin x cos²xdx is x sin 4x {A} 16 64 sin 4x x {B} 16 16 O A B O - sin³2x 48 sin³2x 48 +C + C…

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Q: 8. Find all values of 0, if 0 is in the interval [0°, 360°) and cos 0 V3 2

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Q: Prove that : n/2 1 dx (a'sin'x + b’cos'x 4a°b³

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Q: 5) Determine the following indefinite integral cos (3x) (1 – sin(3x))(2 – sin(3x)) -dx

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Q: 0.5 COS T - In(4 sin z) dz. J0.1

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Q: The integral (cos(3x)) sin(3x)dx under the change of variable u = cos3xbecomes v uz du. -9/29 Select…

A: Given,