Let a, b> 0, then the integral x2a +1(1-x²)bdx, after changing variables y = 1– x will be expressed as.

Q: (b) Use the Fundamental Theorem of Calculus to evaluate the integral .8 So da dx x-6 x+8

A: Need to use the fundamental theorem of calculus to solve this integral.

Q: Evaluate the improper integral 2x dx, if it exists. (x² + 81)2

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Q: Use symmetry to evaluate the following integral. 2 ja (2- |x|) dx - 2

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Q: Evaluate the integral , f(x) dx, where 3 — х3, х 1

A: Given a piecewise function We need to find integral from x=-2 to 3

Q: Find what value of c does ∫∞7c/x^3dx=1 (integral is from 7 to infinity)

A: find what value of 'c' does∫7∞cx3 dx = 1taking integration on left sidec x-3+1-3+17∞   = 1c -2(x2)7∞…

Q: Use the Simpson's Rule to evaluate the integral fx sin(x)dx. W/H N/" N /H 02/ O

A: We have to evaluate the integral ∫0πxsinxdx. Here, a=0 and b=π Let n=2 Hence, Δx=b−an=π−02=π2.…

Q: To evaluate the integral f(x)dx, , the substitution u='g(x) is used. Thus if g(x) = x+4 then dx =

A:

Q: Evaluate the integral dx (2x + 11) by making the appropriate substitution: u =

A:

Q: Use the specified substitution to find or evaluate the integral.

A: To evaluate the given integral using the specified substitution.

Q: Evaluate the integral by making the given substitution. dx u = 7x + 8 (7x + 8) '

A:

Q: Evaluate the integral 7ª S* (y cos x + 4) dydæ.

A: The detailed solution is given below Thank you

Q: Evaluate the integral. (Use C for the constant of i 4x2 + 3x + 4 dx (x² + 1)2

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Q: Evaluate the double integral. 8ху? dA, D is enclosed by x = 0 and x = 4 – y2

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Q: Evaluate the integral. (2 + 2y)? dy

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Q: Express the integral Js(x) dx in terms of Jo, J1 and J2.

A: Bessels function identity:  Jn+1(x)=2nxJn(x)-Jn-1(x) Given: ∫J5(x)dx

Q: 2 Væ? – 1 -dx. x² 1

A: Here , we will solve the integral using substitution form .

Q: Find the integral. J 3 x√9 - (In x)² + C dx

A:

Q: Evaluate the integral f 2x (x2 + 5)" dx using the substitution 16 u = x + 5. 16 2x (x² + 5)" dx = +C

A: Solve integration

Q: Identify the type of generalized integral +00 fx'e* dx- and in continuity (if converging) to be…

A: According to question,Given Integral,Let I= ∫0+∞ x2ex3 dx

Q: Evaluate the integral by making the given substitution.   ∫ x2  √( x3  + 1)    dx , u =  x3   +1

A: The integral formula to be used are: ∫xn dx=xn+1n+1+C, where C is an integral constant. The integral…

Q: Use quadrature formula [f(x) dx = c f(0)+c₁ f(x) to approximate the value of the integral [x² In x…

A:

Q: Use the substitution technique to determine the following integrals:

A: Important integrals for the question: ∫sec2 x dx = tan x + c ∫un du = un+1/(n+1) + c Other important…

Q: Evaluate the following integral x² + y>dydx.

A:

Q: Evaluate the integral. (Use C fo ( 8x2 + 7x + 8 dx (x² + 1)2

A: Given integral is, ∫8x2+7x+8x2+12dx

Q: Evaluate the integral. a dx a > 0 (a² + x²33/2'

A:

Q: Evaluate the integral using an appropriate substitution. dx = +C x√√8-xd

A:

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

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Q: Use the value of the first integral I to evaluate the two given integrals. 3 | - 27 (x3 - 6x) dx =…

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Q: Apply Green's Theoren to evaluate the integral. C6x+y) dx + (3xy"+4y) dy ter

A: Given integration is ∮c(6x+y3)dx+(3xy2+4y)dy C : Any simple curve in the plane for which Green's…

Q: Evaluate the integral f 2x (x² + 3)° dx using the substitution u = x2 + 3.

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Q: Estimate the value of the integral 4- x²y2 dA, S = {(x,y)| x² + y? 0} |

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Q: Evaluate the integral using the given substitution. dx ,u = 6x + 1

A:

Q: Use the specified substitution to find or evaluate the integral.

A: Consider the provided integral, ∫01 dx23-xx+1 Evaluate the provided integral. ∫01…

Q: Evaluate the integral using the given substitution. Sx4x5 -s;² dx , u =x5 -8 * - 8) + C

A: We have to solve given integral by substitution.

Q: Evaluate the integral 35 dz 4 by making the substitution u = x* – 6. +C

A: integration

Q: Consider two integrals: (b) S, x³dx Where a, B> 0 are both positive real numbers. Given that a < B,…

A: In the question we have to determine which integral is larger.

Q: Evaluate the integral f 2x (x² + 2)° dx using the substitution u = x? + 2. | 20(a* + 2)" de +C

A: The given integral is: I=∫2xx2+26 dx

Q: Evaluate the integral f 2x (x² +1)** dx using the substitution 14 u = x² + 1. | 20 (a² + 1)“ dx +C

A: The given integral is: I=∫2x(x2+1)14dx Substitute u=x2+1 After differentiating,…

Q: Evaluate the improper iterated integral. • 1/x9 y dy dx Jo

A: In this question we have to evaluate the integral.

Q: Evaluate the integral f 2x (x2 +7)° dx using the substitution u = x² + 7. 2x (x? + 7)° dx +C

A: Solution:

Q: Evaluate the integral, 00 xn+1e-a²x* Jn(bx)dx, a > 0, b> 0,

A: Bessel's function of first kind of order n, denoted by Jnx is defined as Jnx=∑r=0∞-1n22rr! γn+r+1x2r…

Q: 1 dz c+x2 c)

A:

Q: Use the Trapezoidal Rule to evaluate the integral S |r| x²dx O 2 O -2 O 1 -1

A: Given Intergral ∫-11xx2dx. We have to find the value of it by using the Trapezoidal Rule. The…

Q: Evaluate the integral f 9(√x² + y²) dxdy.

A:

Q: Evaluate the integral by making the given substitution. (5x)tan(5x)dx, u = 5x + C

A:

Q: 3. Which integral should be evaluated using substitution? 9dx (b) dx 1+x2 1+9x2 (a)

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Q: Find the value of P which the integral convesg'es,absolubely Conditionally Sin Vz) dx .P

A:

Q: Evaluate the following integrals.

A: ∫x1-9x2 dx Put, 1-9x2=t⇒0-92x=dtdx⇒-18x=dtdx⇒-18x dx=dt⇒x dx=-dt18

Q: In the indefinite integral J 3x²e* dx, the integrand is A 3r © 3x2 D x³+C B.

A: The given integral,    ∫3x2ex3dx

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.