Show that {zn}nEN is Cauchy, where z, = sint dt. 1+t

Q: t' sin(2t n(21*)dt Evaluate

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Q: r(4) (arcsin(t), arcos(1),0

A: Given: rt=arcsint, arccost, 0   As we know that; ddxarcsinx=11-x2ddxarccosx=-11-x2

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…

Q: Integrate the function. S dt o. O. V6-R+ sin'1 6t + C 6t o. + C 612 o. 6t 16-12

A: We can find the integral as below.

Q: Find/ S dy

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Q: 6) If y=x² +1+ f dt , prove that y"= 2 - x2

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Q: y = tan (5t - 21 +1)

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Q: 3 Let r(t) = · (1 ²³₁,e², Vt-2). t-1 √ F(t)dt = { Evaluate the definite integral.

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Q: L L'FCE)]> F(4) = 4 sin #t

A: The solution is given as

Q: Prove that dl = (Vu) x (Vv) - do.

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Q: Use the second fundamental theorem calculus along with the chain rule to find sint dt dx sinx 2x…

A: We have to find          ddx∫1xsint dt using the second fundamental theorem calculus along with the…

Q: Find o r(t)dt if r(t) = (cos t, sin t, 3t²). %3D O A.O B. (0, 2, ³) OC (0,0, x³) O D. (0, 2, 7) O E.…

A: We have to choose the correct answer from the given option

Q: cos ax -dx = te' 2 х* +1 ле",а 20.

A: In this context, prove ∫-∞∞cos axx2+1dx=πe-a,a≥0. Consider, ∫cfzdz=∫-∞∞cos…

Q: .3 1+ at dt

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Q: What is dy/dt of the given? Show both the simplified and unsumplified answer.

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Q: prove that | Sinsy | dy лу ∞

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Q: (p;-q,)? 2. Prove that KLD(p||q) = J6 t Eisisn pitt(ai-p) dt. %3D

A: Given: To show the given statement as follows: KLD (p|| q) = ∫01t ∑1 ≤i ≤ n pi-qi2pi+i( qi- pi) dt

Q: linearise R = AT + BT

A: To linearise the below equation. R=AT+BT2

Q: Find dx (e3t + 1)dt. -cos(3rx)

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Q: Compute sin (3r)dr (25z4+1)2 -

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Q: Prove that 3x +8 Viv = dr dy: %3D where R [-1,1] x [0, 2).

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Q: Show that fi (x) = x, f2(x) = cos 2x are orthogonal on

A: Given that, f1(x)= x, f2(x) =cos2x and interval is [0, 90o] Wehave to find that the functions are…

Q: Show that Sx cos*x dx = (x² + x sin 2x +cos 2x) + c

A: Given: ∫xcos2x dx=14x2+xsin2x+12cos2x+C.