(2 if 0 s x s 5 if 5

Q: Ex. Find the general solution to dy 1 -2.x dx X -

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Q: 0,-T <x<0 tsen x, 0sx < T 9. f(x) =

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Q: 3.b) Evaluate the DE * 3x(xy – 2) dx + (x³ + 2y)dy = 0

A: Given that: 3x(xy-2)dx+(x3+2y)dy=0

Q: and Suppose z is a function of a and y, and tan Vy2 + x² = z*e6y. Solve for dx dy

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Q: Solve the following initial value problem for y as a function of x dy 3+ /39 Vx² - 2- 25 = 1, x> 5,…

A: Here we have to use the following formula 1)  ∫dxx2-25=ln x+x2-25+c2)  lnab=ln(a)-ln(b)

Q: Let y=(f(x)+5x^2)^4 and suppose f(−1)=−4 and dy/dx=3 when x=−1. Find f'(−1).

A: Given y=fx+5x24 and suppose f-1=-4 and dydx=3 whenx=-1 , Find f'-1

Q: 18 12 | f(2)dz – f(2)de = f(x)dx | f(x)dx 4 where a = and b =

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Q: 4. If o f (x) dx = 5 and S f (x) dx = 6, find So f (x) dx.

A: To find  Integral o to 4 f(x) dx from the given information.

Q: -3 f(x)dx = 8, | f(x)dx = 1, Let f(x)dx = 4. -5 1 -5 Find f(x)dx

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Q: If S 3 f(x) dx = -9. 0. Then f f(In(x)) dx 2x 4 2 O None

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Q: 3. fy dy + 10g = e + dt +7 -2t

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Q: 1. Find -for each of the following: dx e) y =x* Inx f) In(x + y) = e*=y g) y = x2r-5

A: Here we have to find dy/dx .

Q: 2. Find by implicit differentiation. dy dx 4 (a) x +xy +y = 5 2 2 3 =D5 3 2. (b) x- xy + y 3. =D1

A: Recall: An equation of the form f(x ,y)=0 , in which y cannot be expressed directly in terms of…

Q: 1- Find for the function yt + xy2 – 2 = 0 for x = 1, and y = 1. dx

A: We'll answer the first question since the exact one wasn't specified.  Please submit a new question…

Q: Given そ< fcx)59 for S5 xs6 estimate the volue of s f cx) dx 5. S. foo dx s

A: Here we will have to find out the upper and lower limits of the integration for the given limit of…

Q: 18 12 f(x) dx – | f(x) dx = f(x) dx - where a = and b =

A: Find value of a and b.

Q: 2. gxc2x2+3 dx Selast & du 니 get x in terms of u ut 2x2 +3 4-3 = ㄹ x² 4-3 x = 이들 4-3 z soke har x…

A: Given ∫x2x2+34dx

Q: 2xージ f (x) : 33 Vx2 T sin 2x + Ve2x + 2x –- In e, x # 1 1-x (1–3t)5(t³-8) g(t). t # 2 2-t

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Q: dy Solve the initial-value problem for y as a function of x. (49 – x²). dx = 1, y(0) = 4 Y =

A: We will solve this integration using trigonometric substitution as follows

Q: Q3: van fy = xy-1+ x.ln(x) J f(x,y) = x' + y* if it was

A: To find the partial derivative

Q: (-4x-12) dx For J x+10)³ (x-4)³ F(x) – 1. What is F(3)? -

A: Given: ∫-4x-12x+103x-43dx=Fx-1   ...(1)

Q: 1² + 3x + 4 If t = (r² – 1) what is dt/dx?

A: We have to find derivative

Q: 4. If ( Inx dx = f(x)[In(x) + 1] + C, then f(1) = A 1 B D -1

A: Solution

Q: 27. Select the correct one SEF (x)dx S% \f(x)\dx, | - Sif(x)dx < S%\f(x) |dx,

A: Given- Consider the all options given below, To find-  The correct one in the above options.

Q: d S[e-In(2t+1)]dr. find g(x) - dx A e-In(2x+1) 2 B (2x + 1)² (2x + 1)² (2x+1)² If g(x) = D

A: We have to find out the derivative

Q: 1 10x Evaluate f dz S dx e dy. (10e-10)/3 10e/3 10/3 10e-10 10

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Q: So(VE + vZ)° dx = f° xdx - (Vã + vZ)dx • f;(Vã + v2)dx 6. So x(x² + 1)dx = - S x*dx + f, xdx 7. S'…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: 2. Suppose that . f(x)dx = 9 and S g(x)dx = 4. Find the exact value of 2f(x)dx.

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Q: d Show that In dx 1 (a) x+2 x* +3x +2 1 (b) Hence find the exact value of dx in simplest form. r*…

A: iii.a. Quotient rule of differentiation states that ddxfg=f'g-g'fg2 Chain rule of differentiation…

Q: 2 If "IF(x)dx=6 and frox)dx=7.50mm [rodr then f(x) dx = 5

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Q: 18 13 f(x) dz - f(x) dx = f(x) dx where a = and b =

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Q: 4. ydx + x(Inx - Iny - 1) dy = 0, when x = 1, y = e

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Q: -11 If " f(x)dx = 1 ; S,f()dx = 22 find f" fx)dx. Eiles

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Q: dy Solve the initial – value problem: = y² given that f(0) = -1 dx %3D

A:

Q: The

A: This is Linear Differential equation in x. We use method of separation of variables.

Q: 4. (a) Find (b) Hence evaluate d dx √1+x² X S (1+x²)³/2 dx.

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Q: - 3-z2-y 2. Evaluate L* L z dz dy dx -5+x²+y²

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Q: 4-compos dx dy 5-find dry if f(x, y) = xe xe 6- find (x²³/2)

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Q: 5. Let y=(f(x)+5x^2)^4 and suppose f(−1)=−4 and dy/dx=3 when x=−1. Find f'(−1).

A: Given , y = (f(x) + 5x2)4 and also given that ,f-1) = f , dydx =3 when x=-1 , we have to find f'(-1)

Q: .5 If f(x)dx 4 and | f(x)dx =7, then f(x)dx = %3D

A: Answer is below

Q: 42. a. Find df /dx if f(x) = 2 In. dt. b. Find f(0).

A: Solution is given.

Q: If S x²+x-2 (x2-1)(2x-x²) dx AS + Bf f(x) dx + D f g(x) dx, then x dx can be resolved into A ...

A:

Q: 16 Suppose we know that (3fx)-2)dr-6 4 and that Sxf (x²) dx =8. ise this information and properties…

A: We have to find ∫04fxdx

Q: 5. ydx - (x +3y')dy = 0 Ans. x = 6y In C

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Q: Use Green's function to solve: y"-4y' + 3y e, y(0) = 1, y'(0) 1.

A: Let y1x and y2x be the homogeneous solution of the initial value problem y''+pxy'+qxy=fx, yx0=y0 and…

Q: Solve dx/dt+3x=e^-2t given that x=4 when t=0

A: dxdt+3x=e-2t Comparing with dxdt+Px=Q, we get P=3  &amp;  Q=e-2t Now integrating factor is, I.F=e∫P…

Q: 5 If [F(x) dx f(x) dx = 7.4 and 1.°F(x) dx f(x) dx = 5.7, find [²1 f(x) dx.

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Q: -1 x-x2 dx -8 2도 1423 1521 22 20 Option 1 Option 2 1425 1523 22 20 OOption 3 Option 4 None of the…

A: Given problem:-

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.