Given the taylor series for ex= 1+x+1/2!x2+1/3!x3+...+1/n!xn Find the taylor series at 0 for e2x^2

Q: An swer the po llowrng lim f (x) xータ-2f 2. Im t(x) Inn

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A: ∫e-x3e2x-2e-2xdx Solution :∫e-x3e2x-2e-2xdx ∫e-xex3e2x2-2dx 12∫132e2x2-1dx let ,32e2x=t32e2x2dx=dtdx...

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Q: Determine the Laplace Transform of the functions shown and simplify.

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A: Given:  1) limx→432x-3+5 Plug in the value x=4           =324-3+5             =38-3+5              =...

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Q: One of the following is an improper integral. \. xV16 – In x ~In 3 dx dx In 2 V4 – ex xV16 – In x -I...

A: We have to find the value of improper integral.

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Q: 3. Given n > 1, let S = E-1: Show that S, is not k=1 _k° an integer.

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Q: Evaluate the integral S/5+12x² – 4xª xdx (5+ 12x2 - 4x9) +C (2x2 - 3)W5+12x? - 4xª + 14Arcsin - 4xª)...

A: We can evalaute the integration as follows:-

Q: z° – 6z' + 4z – 2 dz |

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Q: One of the following is an improper integral. - In 3 e* dx JIn 2 V4 – ex dx xV16 – In x - dx · In 2 ...

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A: Solution of given question is written in step 2

Q: 4x3 %3D 1. mtegrate. de - aya dx 4x3 du = 4x° dx %3D Jxt -2 xp: こ -112 du te 2(x*- 2) ° +C %3D %3D

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Q: Evaluate the integral ex 2e-2x - 3e2x | Arcsin V6e2x 2 A V3 +C ®Arcsin l V3e2x +C B © 2/2e-2x - 3e2x...

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A: topic - derivative of a function

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A: We will use the following concepts:-

Q: -In 3 e* In 2 V4 - e*

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Q: 2. (а, By completing the square, show that 1/2 d.x x2 – x +1 3/3 By factoring x³+1 as a sum of cubes...

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Q: Use the binomial serles to find the Maciaurin serles for the function. 1 f(x) - (1 + x) Σ (x) = %3D

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Q: S 5e¯x- 6e-3x Given: Find: u 5e2x – 6 O 5e-x- 6e-3x O5e-x- 6e-3x O 5-6e-2x

A: Given:  ∫ex5e-x-6e-3x3 dx To Find:  To find u substitution.    

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Q: By inspection, find the following integral 1 dx 2(5 – 3x)

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Q: Evaluate the integral cot 2x V cot? 2x - 15 13. -순사 Csc 2x +C -Arcsec 4 А. Arcsec Csc 2x +C 4 1 Arcs...

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Q: k! Σ 4k k=0 10

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Q: sec3(In x) + sec²(In x) tan(In x) dx Given: 2x Find: du V sec (In x) + sec²(In x) dx tan(ın x) dx ta...

A: On solving this we will get