Evaluate the real integral J dx by complex variable methods. (x² + 4)(x² + 9)

Q: Evaluate the definite integral.

A: ∫02 14-x2 dx

Q: 2. Express the general solution in terms of Bessel functions: 4x²y" – 4xy/ + (16x² + 3)y = 0

A:

Q: 3. Use the Fundamental Theorem of Calculus to evaluate the definite integral 16 x Vx dx 4

A: Given: ∫416xxdx Simplifing the given integral∫416xxdx=∫416x(x12)dx                 =∫416x1+12dx…

Q: Evaluate the integral. X- 8 dx x2 - 7x + 10 Jo

A:

Q: (2) Evaluate the definite Integral √² (6x +9) dx

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Q: Solve the integral. r5 x° + 3 1. x2(x² + 2x + 1)

A:

Q: Evaluate the definite integral.

A:

Q: Evaluate the integral. х — 1 dx x2 - 7x - – 8

A:

Q: Evaluate the definite integrals a) | (15x2 + 6x - 7) dx = %3D ... ... b) (x- 3)2 dx = %3D ... ...…

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Q: Evaluate the integral. √3/2 x² V1-x² dx

A:

Q: 52 7 -1 dx = V7x – 3

A:

Q: 14. Evaluate the integral 2 Jo dx 2x + 3x + 1

A: ∫0122x2+3x+1dx Factor the denominator  ∫012 (2x+1)(x+1)dx Now apply partial fractions   2…

Q: Evaluate the definite integral. XVx- 2 dx

A: ∫23xx-2dxSubstitute x-2 = tDifferentiate x w.r.t tdx = dt Changing limit :i) If x = 2 then t = 0ii)…

Q: Use the residue theorem to evaluate: o dx 0 x3+1 2n/3

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Q: 1. S xdx x²+6x+13

A:

Q: Evaluate the definite integral. XVx - 2 dx /2

A:

Q: Evaluate the definite integral. | V2z + 3dx 0.

A: Given: The given integral is, I=∫012x+3dx

Q: 4 dx = x² + 1

A: Given that, ∫-114x2+1dx=4[tan-1x]-11 =4[tan-11-tan-1-1]=4[π4-(-π4)]=4[π4+π4]=4×π2=2π

Q: Evaluate the definite integral. dx 6x – 3 1

A: *to find:- to evaluate the given integral. *concept:- logarithmic property. log(m) - log(n) =…

Q: Evaluate the definite integral. 3. Vx2 dx

A:

Q: evaluate the definite integral.

A: Here given Use the standard result of integration as

Q: Evaluate the definite integral. 4 (x2 - 3) dx -4

A:

Q: Determine the integral solutions for which x and y are positive. 2x + 5y = 17

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Q: the Fundamental Theorem of Ca -5 (x³ – 9x)dx. v3

A: Here we have to use fundamental theorem of calculus to find the exact value of  ∫05(x3-9x)dx…

Q: Show that 3x2 +6x +2 = 3 (x+1) - by completing the square. Hence, evaluate the 2 integral dx. V3x? +…

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Q: Evaluate the integral. X - 4 dx Jo x2 - 5x + 6

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Q: Evaluate the integral f 2x (x² + 5) dx using the substitution 13 u = x +5. 2x (x² + 5)*° d =| +C

A: I =∫ 2 x (x2+5)13 dx  u = x2+5du = 2x dxI =∫ u13  du We know that,∫ xn  dx =xn+1n+1+C

Q: -3 x²+4x+3 dx 13.J, 33+2x2+x

A:

Q: 4. Evaluate the following improper integrals using Cauchy's Residue Theorem: (Convert them into a…

A: Note: We are solving the first two questions

Q: Evaluate the definite integral 3 2 (а). (Зx — х + 6) dx [3x(x² – 1) dx - (b). 1 5 -dx (x² +10) (c).

A: (a) Given integral is ∫-13(3x2-x+6)dx…

Q: Evaluate the integral. ∫ (x2 − 3x) dx ______________C+

A:

Q: dx 1. -1 V1 – 2x – x²

A: Explained below

Q: 5 4x2 +7 dx. J2

A: Given query is to find the value of the integral.

Q: Find the complete integrals of the equations (i) px - 4q°x² + 6x²z=2 =0

A: As per our guidelines we can answer only first question. If you want another question kindly repost…

Q: Use the Fundamental Theorem to calculate the definite integrals. 2 1 dx x+2 -1

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Q: Evaluate the definite integrals using the Fundamental Theoren 3. (a) (x+2)(x-2)dx (b) (4ex ) dx 1/4…

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Q: Solve the separable equation using the integral factor  2xy + 6x + (x²-4)y¹ = 0

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Q: Evaluate the definite integral: 1/4 dx V1-22

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Q: Find al integral values of 'a' for which the equation (x - a)(x - 10)+1=0 has integral roots.

A: Solve for the value of a

Q: Evaluate the definite integral. dx 3x + 6 0.

A: This definite integral can be solved by using the Substitution method as explained in step 2.

Q: Evaluate the integral (2х — 8)dx - x2 – 6x + 18

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Q: Evaluate the definite integral. Watch It dx 1 + 3x? 3.

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Q: Evaluate the definite integral. 3 (x² – 2)2 dx

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Q: Evaluate the definite integral. V3/2 1 dx 1+4x²

A: We use Trigonometry substitution method to solve this integral

Q: Evaluate the integral (2x + 4) (2² + 4x + 2) dz by making the substitution = x² + 4x + 2. Submit…

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Q: Find or evaluate the integral by completing the square

A:

Q: Evaluate the definite integral. (x2 - 2)2 dx

A: To find the integration

Q: Evaluate the definite integral. 5x2 + 1 dx = x2

A:

Roots

The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form of a + bi.