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: 1. Which of the following is incorrect? a. b. [(f(x) · g(x)) dx = [ f(x) dx . [ g(x) dx [(f(x) +…

A:

Q: dx =? (a2 + 1) (x + 2) O A. 1In|22 + 1 + 6 arctan æ + 3 In|r + 2|) +c O B. CIn|z? + 1|+ 6 arctan a +…

A: By using partial fraction method

Q: 7 3 Suppose that f and g are continuous and that f(x)dx = - 3 and g(x)dx = 9. Find (g(x) - f(x)] dx.…

A: Given: ∫37f(x)dx=-3 ; ∫37g(x)dx=9To find the value of ∫37 g(x)-f(x) dx=?

Q: 3. Please prove[x"In(x)] = x¹Jn-1(x) and [x"n(x)]= -x¯¹]n+1(x) dx

A: (.) We have to prove,(.) Bessel's functions  is given by,

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: 24. By Simpson's rule, evaluate Vi – 1/x dx with fiye ordinates

A:

Q: Expand the function f(x) = x sinx, as a Fourier series in the interval - < x < π.

A: Given function is  for                    here               therefore f(x) is an even…

Q: 7. Find A. 6 C. 2 fre f(x) dx, given [*[³f(x) + [3f(x) + 4] dx = 18 B. 10 D. 14 3

A:

Q: + 2x Compute dæ. O 3(z+z²) +C 1 1. I미 피 + C 2 O x +2 ln |æ|+C

A: Given: I=∫1+2xx2dx

Q: 1. (a) Evaluate Jo 0 r* - ¹ (1 − x6) 15 (1+x) ¹5 -dx.

A:

Q: -9 8et dt

A: I=∫−948etdt

Q: ...R . The graph off consists of line segments, as shown in the figure. Evaluate each definite…

A: Given information:    Basic information:

Q: Suppose that g is continuous and 7 that g(x) dx = 3 and 2 10 7 g(x) dx = 17. Find g(x) dx. 2 10

A:

Q: 3 -1 -2 -3 -4 1 2 3 4 5 6 7 8 9 10 Compute each of the following definite integrals: (a) f(#)dr. 10…

A: Given

Q: 8 8 Suppose that f and g are continuous and that f(x) dx = -2 and g(x) dx = 8. Find [[4f(x) 4 OA. 24…

A:

Q: → (+,-) (- -) J (4)

A:

Q: 2. Let g(x) = "(x² – 4x + 6) dx. Is the graph of g concave up or concave down at x 1? Justify your…

A: Explanation of the answer is as follows

Q: Liouville function X(n) is given by formula

A:

Q: Compute 2 /x. F(1+x³) dx -1 given that 9. f(x) dx=6 Your answer should be an integer.

A:

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: Which of the following represents the area between the curves f(x) = -x + 2x + 3 and g(x) = x + 1 on…

A: Given Data: Curve 1: fx=-x2+2x+3 Curve 2: gx=x+1 Interval: [a, b]=[-1, 2] The graph of the given…

Q: dx -= In[F(x)]+C. What is F(3) if C= 0? For (x+x³) (x Tan-x+ 1) A 6 в 3 3 D + + -15 -15 C.

A: Given: ∫dx(x+x3)(xtan-1x+1)=lnF(x)+C    ...........(1)We have to find F3 if C=0

Q: 1. Which of the following relations is FALSE. n+m A) (ab)" =a"b" B) C) (a")" 2. The equation of the…

A: To identify the false relation from give question.

Q: Compute 0 O-1.5 O-1 1.5 01 L₂ f(x -2 f(x) dx.

A:

Q: dx - = In[F(x)]+C. What is F(3) if C= 0? For %3D (x+x³) (x Tan-!x+1) -1 A +

A:

Q: |3 – x|;x 5 10 4. Evaluate the integral f(x)dx geometrically for f(x) = {

A:

Q: 2:10 O 4G+ l al 2 2x4 + 1 Let f(x) then f(x) is x+ 1

A:

Q: If yields ao ln x + a₁ ln x + 1| + + an ln |x +n + c where ai (i = 0, 1,,n) and c are real…

A: Partial fraction decomposition is a technique to represent a rational function into proper rational…

Q: f(x + a) dx = f(1) dæ = | f(x) d.r. -T+a

A: Proving of integral of periodic function

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.