t sin (t®)dt

Q: 13. s = cos* (1 – 2t) 14. s = cot t

A: differentiate both sides with respect to t and solve the question.

Q: Integrate S(cos x)e*+sin xdx B. et-sinx + C A. -e4+sin x + C c. e4+sinx + C D. e-4+sinx +C A В C D

A: Given: I=∫cosxe4+sinxdx  ....(1)   We know that; ∫eax+bdx=1aeax+b+C

Q: Determine 1 dx , 4x2 +1 (i) (ii) x cos x dx .

A: Determine.  (i).  ∫14x2+1dx             ⇒∫14x2+1dx=14∫1x2+14dx  ⇒∫14x2+1dx=14∫1x2+122dx…

Q: Find dt 1- t2 sin (t)

A: Solve the integration

Q: x = 1+ cos t, y = sin² t

A: on computing the derivative of the given curves with respect to t than we will find the dy/dt so

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

A:

Q: c) Use Part 2 of the Fundamental Theorem of Calculus to find the derivatives. d i) EevEdt dx dx J…

A:

Q: 3. y' + (tan t)y = sin t, y(t) = 0

A: Sol

Q: Differentiate.  g(t) = t cosect t - cot t

A: To differentiate:      g(t) = t cosect t - cot t We know that --> (cosec x)' = -(cosec x)(cot x)…

Q: b dx = 1-sin(bx) %3D

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: [ sec2t dt + S sin?t dt + § n'-1dt

A:

Q: f(t) = sin t 0<t 2

A:

Q: T Cos x-sin x Jr x² dx So

A:

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

A: We have to solve given problem:

Q: •T/2 cos“ x dx. OS

A:

Q: i) f si sin 3x cos 3x dx ii) [ cos³x dx

A:

Q: dx I cos (x)

A:

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

A:

Q: (b) I cos cos(√x + 3) dx

A: b Given integral is ∫cosx+3dx. We have to solve the given integral. Use formula…

Q: Differentiate.

A:

Q: Find dg(t)/dt -1 1+ cos t g (1) = %3D sin t

A:

Q: Calculate :| t' sin(t) dt [Hint: At some point in the solution, it might be helpful to write t as t2…

A:

Q: F(1) = [cos (t – ;) + sin (t -))« (- -;) + sin (t –)] u (t – ;) 2

A:

Q: . t'e-t + sinh(t) te2sin3t + tsin5t + 4(-1)² t2 sint + sin (3t + 2)

A:

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

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

Q: mifferentiate with respect to t. y = a cos(t) + e sin(t)

A:

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

A:

Q: X Coy-Ycor I+ Sinz +Siny

A: Given that, f(x,y)=xcosy-ycosx1+sinx+siny The derivative with respect to x is,…

Q: TE/7 b. π/3 Sa sin(t) dt

A:

Q: esin a x b. COs x dx 0.

A:

Q: X = 2 cos t [y = 4 sin t |||

A: Using the arc length for parametric equation Formula: - Consider the curve defined by the parametric…

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: c) Use Part 2 of the Fundamental Theorem of Calculus to find the derivatives. d i) da So evEdt d ii)…

A:

Q: L {t®e-%t} L{e-t cos 2t} L {2e3t sin 4t}

A:

Q: In Exercises 5-24, compute the derivative.

A: cos xsin x=cot x, 1sin x=csc xDerivative formulasddxcot x=-csc2(x), ddxcsc x=-cot x csc x

Q: Differentiate the function F(t) = (ln t)2 sin t

A: Given function is,

Q: sin(x) d Find (S sin (4 t) dt). cos(2)

A: Use the concept of derivative of an Antiderivative.

Q: cos t dt 1 + sin t 37.

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

Q: x = sin(t/2), y = sin(t)

A: answer is in next step

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

A:

Q: xdv S- le2cos(sin-w)

A:

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

A:

Q: dx a > b > 0 a + b cos x

A: Definite Integration

Q: sin't dt + cos?t dt

A:

Q: Differentiate

A: Derivative of a function is the rate of change of the function. It gives the slope of the function…

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.