3 In(1+ V sin²x .limr→0 V1– cos x

Q: Consider the following IVP: y" +y' = -cos x; y(0) = a, y'(0) = b Find the general solution. A y(x) =…

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Q: 2. Find an interval where f,(x) = sin2 x and f2(x) = cos² x are linearly independent. %3D

A: W(π/4) = -sin(π/2) = -1 W(π/6) = -sin(π/3) = - √3/2 Since, Wronskian is not zero for a single value…

Q: 7. Evaluate the integral. J CoS X dx 0 1+sin2x

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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: 7. Differentiate w.r.t x- sin 5x cos 3x

A: Given problem:- Differentiate w.r.t  x sin5xcos3x

Q: ) Evaluate A) /16 /2 cos (167 x5 ) dx dy. Os xs 2 Og y s x*

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Q: (cos 2y – 3x²y²)dx + (cos 2y – 2x sin 2y – 2x³y)dy = 0 |

A: We will find out the required solution.

Q: x2 Vcos(t) +3 at 1. COS 2.

A: Given that, the integral ∫2x2cos t + 3 dt

Q: 5 Evaluate the integral 2x cos(xy)dydx 4 0.

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Q: 11 Evaluate (( sin(y²)dydx where 2: 0<y<1, 0<x<y. 0 x a) 1-cos(x) b) 1-cos(x)) 1 - cos() 2 d) 1.…

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Q: Find Domain and Range бяц tue Ⓒsin (x+3) +2y x²+2y 3x + y² 3) 3x + Sin (3x) +2y+Z² Ⓒ x+2y cos2x…

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Q: 8. lim,0 1-cos x 2x2 1 (a) 1 (b) – (c) 0 (d) ; (e) 1. -

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Q: Evaluate the integral. 7/2 cos 6x dx COS -T/2

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Q: - Find the extreme value/s of f(x) = 4 sin² x cos 2x on (37 3π 5π 22

A: This is a problem related to maxima and minima of the function. Based on the general formula and…

Q: 15. Evaluate f e sin2x cos 2x dx A. 2ecos 2x + C C. 2e sin 2x В. + C 2ecos 2x D. 2esin 2x + C

A: You need to have the idea how to use substitution in integration.

Q: 1. Find Duf at P: f(x,y) = sin(8x – 5y); P (5,8); u =

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Q: 11 Evaluate || sin(y)dydx where 2: 0<y<1, 0<x<y. 0 x a) 1-cos(x) 1 b) (1–cos(x)) e) ,-cos(1) d) 1…

A: This integral can be solved by changing the order of the integral. The order can be changed by…

Q: azz = e2x-y + cos(-3x + 2y) + 3x, дхду Given that az where əx 5x2 – sin (3x) when y = 0, and Z =…

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Q: Prove: ex. a. ſ csch x = In b. S sech x = 2 Arctan(e*) + c

A: a).  ∫cschx dx=∫2ex-e-x dx=∫2ex-1ex dx=∫2exex2-1 dx Put, ex=t⇒ex=dtdx⇒ex dx=dt Therefore, ∫cschx…

Q: 8. Evaluate the double integral. cos 2x.sin y dy dx

A: Solution-

Q: (d) lim In(cos r) エ→0

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Q: 3e'x In cos2x Vxy = 1 y? x+1 7 In (xy)-3sin 2y |- 2y = 0

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Q: 1. Find Duf at P: f(x,y) = sin(8x – 5y); P (5,8)); U =

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

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Q: Evaluate the integral cos"() sin(x) dæ cos'4 using the substitution u = 1 = cos(x). | cos"(x) sin(x)…

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Q: hD (Hint: E =eD and Sinh x: 2 Q3: Prove that 8 = 2Sinh %3D

A: We will prove the following result.

Q: In(x) limx→0* 0 e = 0 cos(re-x)+2

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Q: Evaluate the integral 3 2x cos(xy)dydx 2

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Q: The Wronskian, W, of the functions y1 = sin(2x) and y2 cos(2x) is %3D

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Q: I need solution to this problem.

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Q: i. Jsin xcos 2x - sin 2x cos xdx

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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: 1.) Evaluate In 2 e*2arctan e* J (1+ e2x) sec(2arctan e*)' А.) -dx. ežr(1 – e*r)- (arcsin e 2x) cos²…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 6xsec 3x 2-If y = (tan3x)²² prove that = (tan3x)²= + 2ln(tan3x) %3D dx tan3x

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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: 1. Find for the following functions : dx 4i) y-bmcx* +2y}kos x (ans.: 4x.cos x - 2 In( x² + 2 )sin x…

A: To find the dy/dx of  y=[ln(x²+2)²]cosx

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

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Q: 13. ebx cos x dx exists only when b < 0. lim t→∞

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Q: Q4. Given that: z = y sin(2x) + (1+x²)e* + cos x tan-1(y* + 1) In(y6 + 137)

A: Given that, z=y sin2x+1+x2ex+cosx tan-1y4+1 lny6+137 ∂z/∂x is the partial derivative of z with…

Q: evaluate the Cauchy principal value of the given improper integral. cos 2r dx x² + 1

A: We have to find the Cauchy principal value of the improper integral. First we have to write the…

Q: 6) Find most general antiderivative of the fxn ) f(2) = secx (secx +tane) Secx +tanx, = secx COS X

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Q: If y = (x – a)/2ax – x² + a² sin¬'(÷ prove that y' = 2/2ax – x² a is constant.

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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: 3. If w-f(x.y) and x-r.cose, y-r.sin6 Prove that - A + fy ar

A: Given that w=f(x,y) and x=rcosθ,y=rsinθ Prove that ∂w∂r2+1r2∂w∂θ2=fx2+fy2

Q: -3 cos (2x)+sin(2x) in the form Rsin(2x+a)where R>0 and 0 <a<2n. Express

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Consider the following IVP: y" + y' = -cos x; y(0) = a, y'(0) = b Find the general solution. @ y(x)…

A: We find the general solution of given DE  y"+y'=-cos(x)........(1)

Q: 1. Find Duf at P: f(x,y) = sin(8x – 5y); P (5,8)); u = (i-)

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Q: in(cos(x)) 2 limy→0

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Q: Find the general solution of the DE: dy/dx = sin(7x) cos(5x) + cos(7x) sin(5x).

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