For positive integers n ≥ 1, consider the functions 2 A (a) Show that (b) Show that Sn(x) = ㅠ [ Sn = sin(nx) [" (5(2))² de 1 for all n. da dx = Sn(x) Sm(x) dx = 0 for all n m.

Q: Var(X)=E([X-Hx])² صواب İhi

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: -= In[F(x)]+C. What is f(3) if C= 0? (x+x³) (x Tan-'x+1) A 3 B 6. 3 D V3 6 . + + -1/5

A:

Q: 12 If [" f(x) dx = 19 and ' [₁ f(x) dx [R²F(x) f(x) dx = 4, find the following.

A: We can answer the question as below using the properties of definite integral.

Q: Let f (x) be equal to -x + 1 for x < 0, equal to 1 for 0 < x < 1, equal to –x + 2 for 1 < x.…

A: To evaluate the integral

Q: Find x and y for which the following equation is satisfied: x+2 = - 5+i i-5

A:

Q: dx - - In[F(x)]+C. What is f(/5) ir c = 0? For I (x+x³)(x Tan-'x+ 1) A 3 B D + 3

A:

Q: d Prove that (x") = n x"-1 dx

A:

Q: dx For - = In[F(x)]+C• What is f(/3) if C = 0? (x+x³) (x Tan-'x + 1) -1 - 6. + -1 - -15 -is -is

A: Integrate ∫dx(x+x3)(xtan-1x+1)

Q: Prove that Jn (x) X good dx=1. n

A:

Q: = f(x) = (4x – 2)(x² – 3). F %3D

A:

Q: 7. Prove that d dx [cos ¹ x] = = 1 √1-x² x2

A:

Q: f (x) = { ₂x² (²/²) ² 2=1 Coso f(z) dz= ? 23

A: We have to calculate

Q: Evaluate. 3 Lox f(x) dx, f(x) = (−x + 1, x≤2 1-x² + 6x-9, x>2

A:

Q: 3 a) Given fg(x)dx=5, [[3g(x)– 4h(x)]dx = 27 . i) Find [h(x)dx . ii) Hence, evaluate [4x – 3h(x)]dx.

A:

Q: Show that esin(x²)dx < e– 1.

A:

Q: b Find 60 such that - So (x − 1)dx = b == = 0.

A:

Q: 2 Iƒ ƒn(x) = efn−1(x) for all ne N and fo If d nd {ƒn(x)}. dx

A:

Q: Determine the mean value of f(t) = t2 - 5t + 3 cos (pi t) on the interval [0, 3/2

A: The solution is given below in the next step:

Q: . What is F(/3) if C = 0?

A:

Q: Find the greatest value of F (x)= | (6 . cos t - 2 sin t) dt 5 7/3 in the interval [5/3,7 t/ 4|.

A: Fro the given problem: Fx=∫5π/3x6cost-2sintdt Now solve the above definite integral,…

Q: 3 2x 11-8 n x³ + 1/ 5 where [ ] = G.I.F If lim /2n dx = k, find the value of [1] + 2013.,

A: Solution :

Q: Calculate M6 for f(x) = 5 · In(æ³) over [1, 2].

A: Given:   f(x)=5.ln(x3) over [1,3]

Q: Show that J−1/2(x) = √2/πx co

A: Bessel functions are a family of mathematical functions that arise in many areas of physics and…

Q: Evaluate L{3 (t² – 1) sin (2t) + 7et cosh (4t)}

A: