cos t dt .5 V sin t

Q: sin(r )dt = i

A: We have to evaluate ∫04sin(t2)dt using calculator.

Q: {e* cos x , e* sin vx}

A: Given that: {excosx, exsinx}

Q: Q. Find F(s): e-Tt St cos? 3t dt.

A: Q. The given integration is,∫0tt cos23t dt

Q: 2. f(t) = v1+ sin t + (8t)15 + t? cos 3t

A:

Q: r(t) = (e' cos t)i + (e' sin t)j + 2k %3D

A:

Q: f (t) = v1+ sin t + (8t)15 + t² cos 3t

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Q: Differentiate the function. y = sin t + T cos t

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Q: (a) J x cos x dx

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: Find d(tan x)>dx.

A: dtanxdx

Q: 118 t' sin (2€) dt

A:

Q: Cos t dt 1+ sin t

A:

Q: dt cos?(t) V5 + tan(t)

A:

Q: ∫ t2 sin(t) dt

A: Explained below...

Q: Differentiate with respect to t. y = c cos(t) + t? sin(t) y' =

A:

Q: e-t sin(v3t) dt (a) L

A: We have to solve given problem:

Q: TT (sin x T (1 + cos x) dy dx Jo 0. 0.

A: Evaluate the given integral as follows.

Q: •T/2 cos“ x dx. OS

A:

Q: r(t) = (tan(5t))i+t cos(t)j – Vtk

A: Given: rt=tan-15ti+tcostj-tk

Q: Differentiate with respect to t. y = a cos(t) + e sin(t) y' =

A: Given : y = acost+t2sintt

Q: cos t (y sin t + 1) y' with y5) sin? t = 1

A:

Q: dx I cos (x)

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Q: dt f(x) = 1+ sin? t

A:

Q: { 0, sin t, ーTくt<0, 0 <t < T. f (x) =

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Q: y(1) = yo + A cos(@t + p)

A:

Q: a) sin(s+t) = b) tan(s+t) = %3D c) s+t is in the second quadrant

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Q: L{S t? cos t dt}

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the 7th question.  Please…

Q: Sot.e-3t. sin 2t . dt f(t)

A: Take the integral: integral e^(-3 t) t sin(2 t) dtFor the integrand e^(-3 t) t sin(2 t), integrate…

Q: f' (t) = sin 9t - cos f(t) = O %3D

A: We have to find all possible functions with the given derivativef'(t)=sin9t-cost7

Q: bn = s(t) sin(nw,t)dt %3D

A: Given :-  bn = ∫-T02T02 st sin nw0t dt

Q: sin(t + a) cos(t + a) 3 dt 8. f(t) = e 2t 0.

A:

Q: T/2 sin d. t COS - dt dx

A:

Q: f(t) = t cos tei + t sin te, + te3, I = {t: 0 < t < T/2}.

A:

Q: f(t) = t'e-" -4t + t sin t.

A: The Laplace transform of the function f(t) is denoted as F(s) or L[f(t)], it is defined as…

Q: b) y = cos Vx

A:

Q: 4. r(t) = cos ti + sin tj + ln cos tk, 0 < t< T/4 ||

A:

Q: t sin (t®)dt

A: The given integral ∫t5sint6dtLet t6=x6t5dt=dxt5dt=dx6

Q: sint In(cos t)dt

A:

Q: Evaluate the integrals

A: To find the integral ∫0π4sect i+tan2t j-tsint kdt

Q: cos t dt 1 + sin t 37.

A: The given integral,37. ∫cos t1+sin tdt

Q: Evaluate (sin t)" cos t dt =sin't + C.

A:

Q: Differentiate. y = c cos t + t2 sin t

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Q: sin't dt + cos?t dt

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Q: Evaluate L{4e¬2t¢3 +t sin t cos t}

A: We need to find Laplace transformation.

Q: cos(5r)ő(1 – 3) dt = e sin(t)6(t – 2) dt =

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Q: d 6. dx sin(t) dt

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Q: 9(y) = [' r*si sin t dt 12

A: To evaluate the integral: ∫2yt2 sint dt Solution: Integration by parts states that: ∫uv dx=u∫v…

Q: dt cos?(t) dr dx du du + ex- at cos²(t) :-7u, ex dr du -7u dr t(0) = 0, x(0) = a, u(0) = e et heR.

A:

Q: Evaluate the integrals

A: Evaluate the integrals

Q: f(t)=sin(-t)- f(t)=1-cos(wt). soo.(10){SI=(1)f

A: We have to find Laplace Transform of given function.