Use mathematical induction to prove that the n-th derivative of (x +n + 1)e" is given by d" (x+1)e = (x +n+1)e* dxn for every positive integer n = 1, 2, 3, ...

Q: 9. 3 If f is continuous and f (u) du = 3, then / (4xf(x²) + f(3x)) dx is equal to

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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: /3 In 3 7e*dx 1+ e 2x

A: Given: ∫0ln337ex1+e2xdx To find: We have to find the value.

Q: 4. If fdx = f(x)[In(x) + 1] + C, then f(1) =

A: We have, ∫lnxx2dx = fxlnx+1+C To evaluate : The value of f1

Q: d Prove that x" Jn (x)] = x" Jn – 1 (x) %3D dx

A: prove

Q: x² e=a* {H„(x)}® dx = (Vn)2*n!(n + }).

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Q: In (ax-1)dx

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Q: ) prove thaE cos ( x) dx =- T e x" + x2 +l 213

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Q: 00 Evaluate "(1– x)e¯* dx.

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Q: Compute 1 (x² + 1)(x² + 4) Lo -dx.

A: Given ∫-∞∞1( x2 + 1 )( x2 + 4 )dx

Q: 7. Compute: a) e³+ i b) sin(i) c) Log(1 + i)

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Q: 5. (i) Show that +C, 1+u (a) du = 2 In u(1+ u) 1 dx du , using the substitution u = x. (b) J x(1+…

A: Given,

Q: Question 3. Prove that L(e^)(«) = * & dt.

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Q: S* (1– x)e* dx

A: To find out the integration.

Q: Then Suppose f(x) dx = 13 and 8 12 4 [ 12 13 and ["fl f(x) dx = f(x) dx = 5.

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Q: e" dx Find (ez +3)(e –5) Find S In(x)x²dx.

A: The first question can be solved using the concept of partial fraction and the second question can…

Q: What is the choice of substitution for S e*(1+e*)(1/2) dx?

A: The given integral,        ∫ex1+ex12 dx

Q: .3 S(3 – e3*)²dx

A:

Q: If a > 0 and [[ 20*+v 20+u° dr dy = 1(e* – 1) , a =?

A: Using polar coordinate evaluate the above integral

Q: (b) Prove that sint -dt > 0 t+1 for all a > 0.

A: Given:- ∫0x sintt+1dt >0

Q: IfI(u) = [log, (1-2u cos x+ u' )dx, then prove that, I(u)+I(-u)-DI(u²).

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Q: -3 (k) [ 6* dx

A: SINGLE REQUEST SINCE YOU HAVE ASKED MULTIPLE QUESTIONS IN, WE WILL BE ANSWERING ONLY THE FIRST…

Q: dN + N = Ntet+2 dt

A: we have to solve first order of differential equation given below

Q: > [2] S 1+e* dx 1-ex

A: The solution is given in the next step.

Q: Find a formula for ƒ¯¹(x) if ƒ(x) = = (²+1)* ƒ-¹(x) = (-¹)'(z) =

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Q: Let f(z) z+ 12 f'(2) =

A: Given that: f(x)=1x+12