* /Q3 e*-(x+1) lim X→0 CoSx-1 %3D -1 1 -2 O O O O

Q: [In(x²)-ex cos(x-1).tan(Tx) 20. lim 2x2-x sin2(1-x) X-1

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Q: Evaluate the integral. 3 1 3y2 cos xy dy dx x/3 5(1 - cos 3) 3(1 - cos 3) cos 3 - 1 (1 - cos 3)

A: Formulae used: ddxxn=nxn-1∫cos(ax)dx=sin(ax)a+C∫sin(x)dx=-cos(x)+Csin(0)=0cos(0)=1

Q: In (**) 1+v1-x²' sech-1 x (A) O sech x

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Q: 6. The improper integral dx converges to /x- 5 '5 Vr-5

A: The given integral is an improper integral of second kind.

Q: Find integral.

A: Given: Let I=∫2x cos x2 dx To Find: To evaluate the given integration.

Q: -Vx+ 1+ In(x) e* lim + sin (x)cos(x)

A: Find the limit

Q: find integral x3 3) S dx /1-x²

A: We have to find the indefinite integral

Q: | 35. Prove the reduction formula for n > 2: cosh" x dx = cosh"- x sinh x + cosh" п-2 хdx п n-1 п п

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Q: 13 1 dx V13 - O convergent O divergent

A: 13-x=z-dx=dz∫113-x3dx we need to find.

Q: x² + x - 1 dx J, x(x2 – 1) is convergent or divergent.

A: We have to solve the integral and check it is convergent or not

Q: 1) y = 2sin (5x) ,2) y = In x'n cosh 5x ,3) xsin=14y = y tan-14x 4) y = ecosh (8x),5) y =- t +1 t…

A: As per our guidelines, we are supposed to answer only one question. Kindly repost other question as…

Q: ,3x lim x→0 CO cosh(Зa)

A: By substituting cosh(3x) in givenimit we can get the required value of Function.

Q: Steb by step

A: We will need integration by parts for this problem. So we assume u and dv and from there find du and…

Q: Only one of these calculations is correct to evaluate the definite integral Inx dx Select one: Inx…

A: Integral test:  Integrals convergent if the associated limit exists and is a finite number (i.e.…

Q: 1. Determine if the infinite integral is convergent or divergent dx x² + 2 - 2

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost the other…

Q: +Jx² +1) Sinh'x=ln(x+ 2. 1. Cosh-x = In(x+

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Q: Only one of these calculations is correct to evaluate the definite integral Inx dx Select one: 1 Inx…

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Q: 4) coth-lx = .In ( 1+x tanh-11 %3D 2 X-1,

A: i am solving question 4 for other question repost

Q: 31. Verify the formulas sinh-x = In |x + Vx² + 1| coshx = In |x +Vx² – 1| (for x > 1)

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Q: Evaluate the integral. (sin Sx cos 5x dx 늘 cos 13x-5 cos 3x + C 26 1 cos 13x - 26 1 sin 13 x + C 26

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Q: よ2 et + 1 lim xー0 |x2 dt = ? 1+ v2 + et 1.

A: We will use DL hospital rule and leibniz rule to solve this question.

Q: Evaluate the integral S cos ³ x + 8 sin x cos³² x + 6s in ² x dx; cosx #0 2 cos²x

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Q: 2. Find the indefinite integral: S (Vã+ - x + - 1) dx 6 .3 X X

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Q: Evaluate the integral cos"() sin(x) dæ cos'4 using the substitution u = 1 = cos(x). | cos"(x) sin(x)…

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Q: 7. Use integration by parts to prove that for any positive integer n, | cos"x dx = 1 cosn-1xx+ n - 1…

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Q: 8 (a) To evaluate the integral J cos xdx by using the reduction formula of Cosine function.

A: We have to use cosine reduction formula.

Q: - Using Greens date $(x2 + y)dx bounding of the 0<x<1 and x' sysx.

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Q: 3.12 Еxpress lim n00 n k3D cos 1+ as an integral. (Be sure to include the lower and upper limits|of…

A: Given: limn→∞ 2n∑k=1ncos1+2kn To convert this into the integral , use the following method

Q: b) Determine whether the improper integral 1+ sin x 1+ x2 converges or diverges.

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Q: O f(3x² + sec² 2x) dx

A: Given query is to find integral.

Q: 8x + 1 10. Evaluate the integral / dx. 4.x2 + 12x + 10 11 (a) In |4.x2 + 12x + 10|| tan-(2x + 3) + C…

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Q: Example Let sinh(2)e f(2) Find the residue at z = 0.

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Q: 1. Prove: a. S csch x = In b. sech x = 2 Arctan(e*) +c ex+1 2. 2e* tanh e* dx 3. S sinh(5x - 1) dx

A: Given : ∫2extanhexdx Let us consider that : ex=u. Therefore, exdx=du. So, ∫2extanhexdx=2∫tanhu.du .…

Q: 9-x2 B: Find the integral S 18-x2-y xydzdydx.

A: We have to calculate the given intregration

Q: 18x 18x lim -dx = lim 9x2 + 1 = xp- t 00 9x2 + 1 The integral diverges 8.

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Q: Evaluate the indefinite integral as an infinite series. | arctan(x2) dx Σ + C n = 0

A: Answer

Q: Show that integral 0 to infinity  x2 e -x2  dx = 1/2 integral 0 to infinity e -x2  dx

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Q: 1 lim t→0+ (뉴 1 Vt2 +t,

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Q: Evaluate the integral. S sin (7x - 10) dx ㅇ cos (7x-10) +C -cos (7x - 10) + C cos (7x - 10) + C 7…

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Q: 2T COs x dx

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Q: Evaluate the integral: sinh 3x csch 3x dx coth 3x "+C 3 coth X +C 3 - coth x +C - 3 - coth 3x +C 3

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Q: 3 Evaluate the definite integral dx. (x- 5)4 -2

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Q: Evaluate the indefinite integral as an infinite series. arctan(x5) dx Σ( |)•c n = 0

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Q: (a) (6) Decide if the following improper integrals are convergent or divergent using Basic…

A: To decide the convergence of the improper integrals.

Q: b) Determine whether the improper integral `1+ sin r -dx 1+x² converges or diverges.

A: Comparison test :  Suppose for x&gt;a  . If converges then also converges . If diverges then…

Q: integral from 0 to pie over 4 of sin^2 (4x) cos^2 (2x) dx

A:

Q: Find 뿐 for the ry – ln(xy) = 1 0 뽈 ㅇ-볼 O 래 ㅇ-픕 following:

A: The given expression is xy-lnxy=1 we have to find dydx use the derivative formulas, ddxx=1…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.