Cosx b) Jo esinx) x+1 dx, using the substitution u = sinx. 13. Find a) J1(x+1) e), (x + 1)e *dx.

Q: What is y' of y = sin¹ (cos x)? O A. y' = ATHE sin + 1-cos² 1 O B. y' = 1 OC. y'= sin z 1-cos² z OD.…

A:

Q: Prove that when y = e 2 Sin 2.x, then 4- d y dy +17y=0 +4. dx? dx

A:

Q: dy satisfies dx? 2 iven by

A:

Q: Use Green s theorem to evaluate the give (y² dx + 8xy? dy)

A: In this question we will use basic of calculus i.e concept of integration. We will use green's…

Q: The linear approximation at z = 0 to sin(8x) is A + Bx where A is: %3D and where Bis:

A: Given : The linear approximation at x=0 to sin(8x) is Ax + B   To Find : Value of A and B

Q: IER, t> 0, u(x,0) sin(87r) cos(90z). u + ru, =x

A: Solve partial differential equation by Lagrange method. Rewriting partial differential equation…

Q: Exercise: Answer the following by finding dy/dx. 5sin-1 # – cosh x (a) y = Vsec4 x + In 7 4 (b) + In…

A:

Q: dy Find for ( In (t), at t = 8. dx (Give an exact answer Use symbolic notatio

A: Here x and y are given in terms of t. So to find derivative of y with respect x we have to find…

Q: Find dy/dz in terms of e and y if arctan(x®y) = ry°. ||

A: See the details solution in below

Q: 2. dx Ix Also, Prove that 22 %3D dxdj = dy dX

A:

Q: d Q 3. Prove that tanx dx sec?x. =

A:

Q: How would you choose u when evaluating x" cos ax dx using inteeration by parts?

A: Given integral is ∫xn cos ax dx

Q: J. In (T/2) cos ey dx dy In (/6)

A:

Q: sin(nz) b) E: on (0, 2x] n-1 AcoAnA

A:

Q: If z=In/x2+y2 +- tan 2 then the numerical value of + is equal to..... dy? dx

A:

Q: Evaluate. d [arctan a – In ¤] dæ 1 1+#? 1 sec2 x sec² ¤ 1-x²

A: As we know that: The sum rule: f±g'=f'±g' The chain rule: fgx'=f'gx·g'x The Power law: ddxxn=nxn-1…

Q: 3. The work done W,to stretch a spring from its natural length to an extension of 2 meters Is given…

A:

Q: dy iven that x = e" cos 3t, y=e" sin3t, show that tan 3t + dx %3D 4

A:

Q: d tanx = sec?x. dx Prove that sec² %3D

A: To prove the equation, we need to simplify the left hand side of the equation. According to the…

Q: What is the difference between dy dy and Ay Use an example y = 4x?+ eX to under stand. 4x2+ ex

A: In the given questioin we have to tell about the difference between dydx,∂y∂x and ∆y∆x. first we…

Q: -s Jatx? 3. 4-/ sinx coss dx |

A:

Q: | 38. Use Fubini's Theorem to evaluate xe" dx dy. 0 Jo

A:

Q: Gren thint sin (A+i6) = xtiy, 28 Prove that, 9父+ =1 snh'B 1) x essec?A -y' sec'A =1

A:

Q: Given tanh(x)+ tanh ¯' (y) = 1, show that, 1- y? 1 – x² dy dx

A: Formula,ddxtanh-1x=11-x2Note- Since multiple questions are asked, I am answering the first question…

Q: Evaluate 2Sin x´dx, by (i) mid-point rule, and (ii) Simpson's 1/3 rule.

A: i)Given:        ∫242sinx2dx   Mid-point rule:       ∫abf(x)dx=∆xfx0+x12+fx1+x22+fx2+x32+...…

Q: dy let y = cos(In(cos(x))), then is dx

A:

Q: In? (In(ax)) dx,a >0 -dr, a a. 2x In(ax)

A: You can solve this problem using the substitution method

Q: Te 2=tnx then prove thot dx 28 Snメー tty? 1+xて

A: We have to use proper calculus applications.

Q: e ln (xle) (2– sec²r)cm#/c°sec³x sec(3A – x) What should be the value of A for -dx to be integrable…

A:

Q: (c) Integrate sin"uelog (cosu) . Where n is 482

A: Integrate the sinnuelog(cosu). Where n=482

Q: a. Show that If z = x+ iy where y 1 b. Evaluate: cosh?(-z) – sinh²(z)

A: Given that z =x+iy For a part , y<0 and we need to prove that | eiz | >1 For b part, we need…

Q: Let I = 3x*cos (x) dx. The approximation of / using the two- formula is:

A:

Q: Suppose f(x, y) = x2 + y? where x = r cos 0 and y = r sin 0. af and Or 1, 0 = 6 Find -when r

A:

Q: 1. Obtain the par hicul ar foluhon 4 DE . 9xy dy a. oy dx = ; uh en x= 2,y=1 b. 4xy dy = 2y (I+ y?)…

A:

Q: Determine the concavity of y = 3 + sin x on [0, 2π]

A: The function y=3+sinx on the interval 0,2π. We have to determine the concavity of this function.

Q: Given that y + x + cos(xy) = 0. Show that dy/dx = ysin(xy)-1 /1-xsin(xy)

A:

Q: 7. Calculate dy/dx at the point indicated: f(0) = (7tan O, cos O), 0=/4

A: Here we will use the concept of parametric equation and Derivative of parametric ewyations

Q: DIfterenti are (뉴) de Ipx tan x -In DIfterentiate exy te Yy 1, tnd y"

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

Q: 1)the f a) 3 cosh z b) 0 c) in d) n -dz by using the residues is equal to

A:

Q: Let f(x, y) = 2e sin(4) of д х of dy Check Answer

A:

Q: show that1) d/dz [cosz] = -sin z

A:

Q: dx 5 + 4 sin a

A:

Q: Exercise: Answer the following by finding dy/dx. 5sin-lr cosh x (a) y = Vsect x + In 7 4 (b) + In y…

A:

Q: Verify the rule by differentiating ∫ du/(a2 + u2) = (1/a) arctan (u/a) + C . Let a > 0.

A: Consider the given integral identity. ∫dua2+u2=1aarctanua+C Differentiate the LHS of the equation…

Q: 0.5 To evaluate the integral 1= x e*dx by using trapezoidal rule such that h=0.1. Then n=. 4 3

A: Let a, b be any interval and fx be a smooth curve that takes the value y0, y1, …, yn for x=x0, x1,…

Q: Find where) i ) et u*rany tany = 1 , U= xInx (use chatn reule ) ii) coscxy) - * + e+y) = 2 yz

A:

Q: (4) If z = f(r, y) where, a = e" cos v and y = e" sin v, show that (i) dz az e2u. (ii) az 2 2 dz -2u…

A:

Q: sin(20nnt) ass A: AS(t)ln:0dd En=1 %3D %3D1

A:

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.