4. If dx J x(x+1)· · (x+n) ... yields ao In |a| + a1 ln |x + 1| + ...+an In |æ + n| + c where a; (i = 0, 1, · · · , n) and c are real constants, find An. 1 (a) n! (b) n! 1 (c) n! 1 (d) (-1)"+1 n! 1 (e) (-1)" – n!

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: 10) Suppose that f is continuous and that f(z) dz = 0 and f(z) dz = 5. Find - 6f(x) dx . -3 -3 A) 30…

A: ∫-33f(z) dz = 0  ,    ∫-35 f(z) dz   = 5find -∫356f(x) dx∫abf(x) dx    = ∫acf(x) dx   +∫cb f(x)…

Q: 10 Suppose that " fx) dx and Ax) dx = -5, and "g(x) dx = -1 and g(x) dx = 3. Compute the given…

A: Given,∫04f(x)dx=7∫02f(x)dx=-5∫04g(x)dx=-1∫02g(x)dx=3

Q: b 1+ e2x Evaluate the integral. ex xp 1 + e2x

A: Given that: ∫01ex1+e2xdx

Q: Q3. (30pt) Calculate the integral 2x³ – 4x + 1 - (1– cos x) dx 1+x? a) for n= 10 according Simpson…

A:

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…

A:

Q: 73. Use the definition of |x\, the chain rule, and the fact that (In x) = - for x > 0 to prove that…

A: Given, dlnxdx=1x, x>0 from definition of x x=x ,     x>0 -x ,  x<0  if x>0 then d…

Q: 1. Evaluate the following integral: (1-e-²x) dx (a) analytically;

A: Note:- For this problem all the sub-parts  are different as methods are different. So, we can solve…

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…

A:

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)…

A:

Q: Let p and g be continuous functions in an open interval I and ão E I. Show that if Yı and Y2 are any…

A: If y1, y2 are any two solutions of dydx+p(x)y=g(x),    y(x0)=y0. where p and g are two continuous…

Q: Derive a formula for In x dx where a. nt-1 b. n=-1

A:

Q: 1. Determine if the following improper integrals converges or diverges dx a. (1+r*)'/3 b. cos(x-1)…

A:

Q: 4 dx Vx(1 + x) roper for two reasons: The interval [0, co) is infinite and the integrand h 4 4 dx =…

A: Here we use substitution so simplify the integration.

Q: If In y = 3 sinh x show that dy (a) (1+r2)()² = 9y², dx (b) (1+ x²)- dy da? dy +x- – 9y = 0. da…

A:

Q: - Evaluate the integral dx using Simpson's th rule e* – 1 3 -

A:

Q: (a) Evaluate the integral 9x"e¬X dx for n = 0, 1, 2, and 3. n = 0 9. o((-") - (-1)) n = 1 е | n = 2…

A:

Q: 11. f(t)=t Int -t 12. y = x² Inx 14. y = (ax+ b)' In (ax +b) 16. f(w) = log (w² + 2w + 1) 18. y =x²…

A:

Q: 2x² +10x+9 dx = ? x.(x+3) integralinin eşiti aşağıdakilerden hangisidir? X B) In x+3 A) In(x² +3x)…

A: Consider F(x) = 2x2+10x+9x(x+3)2 We will solve it by partial fraction Let Ax+3+B(x+3)2+Cx =…

Q: 7. Which of these is a non-elementary integral? In(2y² + 1) А. -dy 4y e-x cosh(2x) В. c. [ e-x +…

A: Non-elementary integral for a given elementary function is an  indefinite integral that is, itself,…

Q: Use mathematical induction to prove that the n-th derivative of (x +n + 1)e" is given by d" (x+1)e =…

A: We use mathematical induction. The detailed solution is typesetted and given in detailed as the next…

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 the integral of (1 2x) e^ (-x) dx from x-0 to x=. с. 2 А. 3 в. 4 D. 1 O B

A: ∫ uv dx = u ∫ v dx - ∫ (u' ∫ v dx) dx. ∫ (1+2x)e-xdx = -(1+2x)e-x + ∫2e-xdx ∫ (1+2x)e-xdx =…

Q: 1. Evaluate the integral of x2 (3 - 2x) dx if upper limit = 2 and lower limit = -1. а. 12 b. -12 C.…

A:

Q: In each of the following, find pdf of Y a) Y= X2 and  fX(x) = 1, 0 0 c) Y = eX  and fX(x) = 1/σ2 *…

A: Given: Then:

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: If dr J #(x+1).… (x +n) yields ao In |æ| + aj ln |r + 1| + ... + a, In |r + n| +e where a; (i = 0,…

A: Given, that if the integral   yields, To find an In the given integral, let n=2 and find the value…

Q: 10. Find the integral of dx x²-4 O In() + C x-2 O In() + C 2x x+2 O fIn( *2) + C X+2 O In() + C x+2…

A: *using concept:- 1. Partial fraction. 2. int(dx/x) = ln(x). 3. ln(m) - ln(n) = ln(m/n).

Q: Evaluate the double integral. eln 3 3 7 dx dy In x ey In 3 3 7 dx dy = (Type an exact answer in…

A:

Q: 18 2 6x dx using n = 2, 4, and 8 subintervals 2

A:

Q: | 45. Apply integration to the expansion 00 =E(-1)"x" = 1- x+x – x +.. 1+x n=0 to prove that for -1…

A:

Q: d ** If g(x) = [e² – In(2t + 1)]at , find g" (x) dx A ex- In(2x + 1) 4 ex - (2x + 1)2 B (c) none of…

A:

Q: The general solution for D. E. e'y' – + e' = 0 is - x² +xy O y = e¬* + In ( Inlx|+c x+y-1 O y = -x +…

A:

Q: z'e? dz

A:

Q: 6. Suppose f(2) is analytic in |2| 1, and f(0) = 1, f(0) = 2, compute: %3D (1) h = fe(s+ 1)2d: (2)…

A:

Q: d If g (x) = ª[e' - In(21 + 1)]dt, find g(x) . dx A e-In(2x+1) 4 (В et + (2x + 1)² 2 (C) (2x + 1)² 4…

A: Topic:- application of integration

Q: e 2x Evaluate the integral of e* +1 xp-

A:

Q: 11. The integral of an(In|4x-12 dx is bln|sec(ln(2x)ª)|. What is b? A. 2 В. 2 С. 4 D.

A:

Q: Evaluate the integral: e3+ cosh2x33+cosh2xsinh2x dx (3e)3+ cosh2x A + C 2ln3 (3e)3+ cosh2x +C B 2 +…

A:

Q: Compute the general antiderivative S a"-1 cos(x")dx COS where n > 1 is an integer greater than 1.

A:

Q: ** If g(x) = S [e* - In(2t + 1)]at , find g"(x) dx ex. (2x + 1)2 A ex – In(2x + 1) 4 ex + (2x + 1)2…

A:

Q: メ1+ x€ Lat F Cx, x = F•dr whera yer CIS elosed c is Sagmants from C0,3) to Co,0) than from C3,0) to…

A: Given F=yexyi +xexyjand path is closed

Q: 8. The value of the integral 2[x] 1o sulsV ar 3x-[x] dx, where [] denotes the greatest integer 2[x]…

A:

Q: In Evaluate the integral: Sé e x²+3x-7 x-2 dx

A: Option (b) x22+5x+3ln|x-2|+C

Q: If J f(x) cos(mx)dx = and f(x) = En=1 cos(n²z) n3+1 then m = O 4 O 25 O 16 0 4 9 O 5 O2 03

A:

Q: -X +7 Evaluate the integral: dx (x-2)(x+3) In x - 2|+2 1n x +3 +C 2 In |x – 2| – 2 In |x + 3|+C 2…

A: First of all we will do partial fraction, then we will evaluate the integral.

Q: 17. Find the values of p for which the integral 3-2, dx converges. 3-2p Орe(-1,1) Op > 1 Op 1 Ops1

A:

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

A:

Q: 23. (a) (2n )! [P,(x)]² dx 2 (n!)ª J_, (1 – x*y" dx (b) By making an appropriate change of variable,…

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.