1. f (z) = nz In z 799 II. f (x) = In z I. f (x) = %3D ez

Q: Find the antiderivative of: x*+1 (n+ 1)2 [(n + 1) In x – 1] if n± -1 x* In x dx = z (In x)² + c if n…

A: Find the anti derivative.

Q: 1 4. First, make a substitution and then use integration by parts to evaluate [Hint: Use the fact…

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Q: 2- Find O for: y= In (x+5x) dx

A: Tpoic- Differentiation

Q: Find d if z = log2 y - 6 logs y. In In 64 dz dz dy y In 2 In 5 dy y In 2 In 5 O Option 3 O Option 2…

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Q: Solve the differendialegimaition: (2x+e)dx + (xay co

A: Make some simplification and rearrange the given differential in exact form.

Q: 2 S (x) + x³)dx = ? -1

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Q: d If g(x)=- -S*[e' - In (21+1)]dt, find g(x) Se dx 4 A (2x+1)² 4 B e+ (2x + 1)² Ⓒe* -In (2x+1) 2 D…

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Q: 뿐 for the following: y = ln(e2z) 0 읊 ○ 2 Oezz ○ ez (In 2)

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Q: 10. f (x) = x² + 4x + 4 Opens up/down:. Vertex: Axis of symmetry: Solution(s): y-int: Domain: Range:

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Q: In(x²+x)-In x+eln 4 dx In e(x+1)

A: Given integral, ∫lnx2+x-lnx+eln4ln ex+1dx

Q: Find the antiderivative of: xn+1 [(n + 1) In x – 1] if n+ -1 x" In x dx = (n + 1)² 1 (In x)² + C 2…

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Q: If f(x, y) = e®¤ . In(16y), then fræ is given by: O 144 e9 In(16y) O - 81 e9 ln(16y) O 81 e9 In(16y)…

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Q: 1 Integrate with respect to x. 4+ 25(3x +1)²

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Q: In x Evaluate S 2 x(4-ln² x) O2 ln | 4 - In² x | +C Oln | 4 - ln² x | +C O ln | 4 - In² x | +C Onx…

A: Given ∫lnxx4-ln2xdx

Q: 12. Find the dicections at.which. the directronal.derivative af.f.(hg)..xSin xy.at.(,a).has.…

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Q: IF X has an exponential distribution with the parameter 0 an Hy = 202, where 0 >0 then find µx, a,…

A: Given that X has exponential distribution with parameter θ. μx=2σx21θ=2*1θ2θ=2 Therefore,…

Q: 1. Evaluate S xe0.1% dx In x 2. Evaluate Sdx x3 3. Evaluate S x³(x² – 1)1°dx | 4. Evaluate (2 +…

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Q: 6 1) Find the general antiderivative for f(x) = 1+ x?

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Q: 1. Evaluate the integral |(x² - 5)e*dx 8.

A: Given that: ∫0∞ x2 -5e-3x dx To find: The value of the given integral. Let I = ∫0∞ (x2 -5)e-3x dx…

Q: Find an integrating factor to the linear D. E. In(9 x+16)y+ y =20 9x +16 p(x)= (6 x+ 14) In(18 x…

A: Given the linear D.Eln (9x+16)y'+99x+16y=20we need ti find the integrating factor

Q: 1) Find dy/dt for if : y = log2 (81'n 2)

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

Q: .2 -dædy In (x) ey

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Q: 2. * S arctan x dx 1 In(x²) + C C 1. x arctan x C A x arctan x -In(x² + 1) + C -- B 1 D x arctan x…

A: Solve the given integration

Q: 1. Evaluate 5 S Sx (x + 3x)dydx

A: Given, ∫05∫0x x(x + 3x) dy dx=  ∫05∫0x (x2 + 3x2) dy dx=  ∫05 ∫0x (x2 + 3x2) dydx= ∫05 ∫0x 4x2 dydx

Q: If P(x) = -1/x then the integrating factor is a. e-* 1 b. С. -x d. e-1/x

A: Consider the linear differential equation: dydx+pxy=qx The integrating factor for above linear…

Q: 8. (a) Show that (b) Evaluato 1/3 3 √1+9x² -4x4 dx dx = ln(1 + √2).

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Q: •1 cln(5) 1- S SmS) xy evx ´dydx r

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Q: х — 1 Q8: Find dx. Vx2 + 4x + 5 A -Vx2 + 4x + 5 – 3 ln (V x2 + 4x + 5)+C -1 В x2 + 4x + 5 – 3 sinh…

A: Option (B) is correct answer.

Q: 2 3. Evaluate the integral ||x dx -2 O 2 4

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Q: where the populations x(t) and y(t) are measured in thousands and t in years. Use a numerical solver…

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Q: S de x² –2 x+1 A x2 --x - In Ix + 11 + C C x2 +x - In Lx + 11 + C B x² -x + In Lx + 11 + C x2 +x +…

A: Given ∫x2-2x+1dx. The objective is to evaluate the integral.

Q: In 3 by Pots tien 3) S (4 - x)e"'dx calculate without integre (f Ju

A: Given,

Q: Suppose that L = 27, a2 = 1, and the initial temperature distribution is f(x) = 27 - x for 0 <x< 27.…

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Q: 7 In x (1) Evaluate the definite integral dx.

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Q: If g(x) = - - S^*[e'- In(2t+1)]dt, find g'(x) . dx A ex-In(2x+1) 4 B

A: We can solve this as follows:

Q: e largest interval in wh

A: In this question, the concept of Interval of Unique solution is applied. Given, the Differential…

Q: et Xi- Nco,1) and n=2 T=X,+Xz Show that Tis Sufficiencs y Statistic

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Q: dy y(ln y- In a+1) dx 1 C = -In 1 C = In O Option 2 Option 1 1 C =-In None of the choices O Option 4

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Q: 6. Evaluate the integral e* dx ex + e-x

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Q: (4) The general solution to (x +3)y" +5xy' - y = 0 converges at least on and is y = Co x2+ r6 +...…

A: 4.  We have differential equation x2+3y''+5xy'-y=0  now take a power series y=∑n=0∞cnxn…

Q: 18. Show that Vxª + x and Vx* x→o by showing that they both grow at the same rate as x² as x' grow…

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Q: 19. f(x. y) In(1 - x - y) %3D

A: As per our company guidelines we are supposed to answer only the first question that you have…

Q: If g(x) = — _*[e'-in(21+1)]dt, find g''(x) . - dx 2 A ex_ (2x + 1)2 B e* - In (2x + 1) 4 с ett (2x +…

A: Given:  g(x)=ddx∫-2xet-ln(2t+1)dt To Find: g"(x) Formula to be used: Fundamental theorem of…

Q: g(x) dx converges,

A: Observe the given graph carefully. Note that, in the interval [1,∞) the function g(x) forms a bigger…

Q: Differentiate the function

A: Recall the following facts on differentiation.

Q: 3+ Evaluate the integral: e³+ cosh2x33+ cosh2xsinh2x dx (3e)3+ sinh2x A +C 2+In3 (3e)3 + cosh2x +C…

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Q: Compute f'(1) where f(x)=1/2 e-2x + ln(x2 + 1)

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Q: 5. What is the most general antiderivative of the function S(2) = z - 2, in other words, what is -…

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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.