4. Prove 1 < | V1+ a dx <2 0,

Q: 1 Prove that 3 dx s VI VI

A:

Q: 5) Evaluate the integral: tan 3 x dx

A: Given integral, ∫tan3x dx

Q: 5.Use trapezoidal rule to evaluate f,(x² + 1)7 dx by taking n = 4.

A:

Q: 1 Evaluate the improper integral dx. (x – 3)3/2' 3 -

A:

Q: d Prove In Jax| dx In x for any constant a. dx

A:

Q: Evaluate the integral 2 dx where fx) if -2 s xs 0 16-2 if 0xs3

A: Given function is The integral isThe integral can be written as: Plugging the function and applying…

Q: 35 If fis continuous and the nearest integer f(x) dx 126 , find f(7x) dx . Please round the answer…

A: We have

Q: Q1- Prove that coth x = 2 csch 2x + tanh x

A:

Q: 19. Evaluate the integral Z √(x^2 − 9) / x^3 dx

A: firstly divide the numerator by denominator part by part to simplify the integration

Q: Q1- Prove that coth x = 2 csch 2x + tanh x %3D

A:

Q: 12. Suppose that g is integrable and that g(x)dx = -7 and [8(x)dx = 2. d: g(x)dx

A:

Q: 33 Vx2 f(x) = + Ve2x + 2x – In e, x # 1 TT Sin 2x - 1-x

A:

Q: 4х+2 Evaluate the integral -dx. (Hint: Use Algebra to help further rewrite your (x² +1)(x=1)…

A: First break by using partial fraction then integrate

Q: (a) Evaluate the integral 1 2 (4-)dy dx

A: Evaluate the given integral as follows. ∫01∫024-y2 dy dx=∫01∫024-y2 dy dx=∫014y-y3302 dx=∫018-83…

Q: Find upper and lower bounds for

A:

Q: Prove: ex. a. ſ csch x = In b. S sech x = 2 Arctan(e*) + c

A: a).  ∫cschx dx=∫2ex-e-x dx=∫2ex-1ex dx=∫2exex2-1 dx Put, ex=t⇒ex=dtdx⇒ex dx=dt Therefore, ∫cschx…

Q: 3 Evaluate the integral |x V2x +1 dx ?

A: ∫x22x3+1dx We need to integrate the given function.

Q: Prove that d /dx ( cot -1 x) = - 1/ ( 1+ x 2 )

A: we have to prove that: ddx(cot-1x)=-11+x2

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

A:

Q: O f(3x² + sec² 2x) dx

A: Given query is to find integral.

Q: 1 The approximation of dx by the composite Trapezoidal Rule with n= 2 is +4 O X-

A:

Q: Suppose Fas satisfies f'cx) = fex). Prove that 3fx)sin(3x) + f'cx)cos (3x) Sfewcos(3x) dx = %3D 10

A:

Q: Solve the problem. Suppose that fis continuous and that " H2) dz-0 and S Ka) dz = 4. Find - 60) dx .…

A:

Q: If y = V1-x² sinx, prove that (1–x²) +xy =1– x². dx

A: Given that :- y=1-x2sin-1x Prove that :- 1-x2dydx+xy=1-x2

Q: 2. Evaluate the integral: 2xe-**dx -0

A: topic - integration

Q: If y = x tan-x- In /1+ x² then ¬1x-In V1+ x² dy is dx

A:

Q: 4. Evaluate the improper integral. dx х2 — 1

A: Consider the given integral as Solving the integral

Q: 57 Evaluate the integral |[xy dy dx. 12

A:

Q: 1 uate the improper integral dx. x(In x)3

A:

Q: if p> 1, (diverges; if p <1. 0o dx Prove that f p- 1 хР 1 Hence evaluate: dx. x6

A:

Q: 3. Prove that So 0 1 x + ex -dx = n=1 (−1)n−1(n − 1)! nn

A:

Q: In(x) -dx Evaluate the integral m(2)

A:

Q: Q.4 Using the method of substitution the integral fxer-3 dx

A:

Q: 2 Evaluate the improper integral 4x e dx, if exists. 8.

A: Given that,           ⇒           ∫-∞2e4x dx   ,   if exist             ⇒         find the…

Q: Q2: - If y = 3sec3t and x = etan3t prove that: - dy (In3)2 + dx y 2logy x %3D

A: we have ,  y=3sec3t and x=etan3t we have to prove that , dydx=(ln3)2·yx·logyx…

Q: 2. Evaluate dy/dx for y= x In x 1+inx

A:

Q: -3 evaluate the integral (x2 – x + 2) dx using its definition.

A:

Q: Compute the definite integral 2 dx Vx² + 4

A: First, we need to compute ∫xx2+4dx

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: (b) Evaluate the integral | xe * dx. (1- x)?

A: Given : The integral is ∫ xe-x (1-x)2 dx .

Q: 2. S*3*+2 dx x+2

A:

Q: 4. Prove that ах x" ea" dx xn- п-1 ‚ax dx а а

A:

Q: If u(x, y) = 2 tan-1 2 and v= In (x² + y?) , prove that: a2v = 0 дхду dx2

A:

Q: 13. Evaluate the integral dx x – 1

A:

Q: 15. Evaluate the integral 10 ·dx (x – 1)(x² + 9)

A:

Q: 5. prove that d (-x) (+ x) = X dx dx d² dx² - x² +1

A:

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.