In 2 ety dy dx In 3 1-

Q: 1 Prove that 3 dx s VI VI

A:

Q: 59. Show that for any number a > 1 -Ina In x dx + e dy = a In a,

A: Given: ∫1alnxdx+∫0lnaeydy=alna

Q: In(x +3)dx %3D 0.8392 1.1999

A: By Romberg integration Oh4=134R2,1-R1,1 Where fx=lnx2+3 a=0 and b=1 h=b-a2 For R1,1 and R2,1 use…

Q: Evaluate the integrals in Exercises

A: ∫-ln20cosh2x2dx∫0ln104sinh2x2dx

Q: 4. Evaluate the integral. S tan (2x) sec°(2x) dx

A:

Q: Integrate sin" uelog(cosu) . Where n=666

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Q: Evaluate the integral : S 8г2 +10х—6 (х—1) (х+1)2 dx = |

A: The given integral is ∫8x2+10x-6x-1x+12dx.

Q: Obtain a-(In x)? dx.

A: Given,  ∫x-5(lnx)2dx

Q: If In = + a?)" dx, show that - (x(x² + a²)" + 21na²In 1]. 2n + 1 1 In

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Q: Evaluate O O In 9 3 2 - In 2 In 2 x² 2 O tan¹ (2) tan¹(1) O ln 2 0 1 3 9 dx. x³ + 1 -

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Q: d Prove In Jax| dx In x for any constant a. dx

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Q: The integral 2n / 14xe dx can be done with the substitution u - and du= dx.

A: Step 3Given,2π∫14xe-x2dx

Q: S Le+6)dx (x+2)*

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Q: 18. y = (In 0)" > dy de = T(In 0)-1) () T(In 6)(-1) %3D

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Q: 2 Find つえdy 2 こ( beny s)

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Q: - (In /3) /4 4 e4x dx 1 + e8x

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Q: Integrate 2 In(11æ) -dx . O a) In (11æ)² + C 2 O b) , In (x)² '2백 + C 11 c) In (11æ)² + C 2x d) None…

A: Given query is to eveli the integral.

Q: IN SA Coefticients 4 Linear in two vanabtes: (x-34+4) dx+2(x-ソ-2)dy

A: In the question it is asked to solve the equation using coefficients linear in two variables:

Q: Evaluate the following integral. 4 3е ху dz dx dy -111

A: Given: ∫-14∫13∫1ex2yzdz dx dy

Q: In(3) e Ver +1 dx In(2)

A:

Q: Show dz 4 In= 는I미페 - for a<0. x <

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Q: In x /1+ (In x)² 6 S -dx

A: Given query is to eveluate the integral.

Q: If r = (3 cosh 2t)i+ (3 sinh 2t)j+ 6tk and t = In 2 then اخترأحد الخيارات

A:

Q: Find * In(x² + 2x + 2) dx (x + 1)² In|1 + (x + 1)²| x +1 In|1 + (x + 1)2| x + 1 + 2tan-(x + 1) +c…

A: I am attaching image so that you understand each and every step.

Q: Evaluate the double integral 2. dy dx 4. 31x

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Q: In 8) * dx Integrate and evaluate the following

A: we have to integrate the following

Q: Evaluate the integral. 7- sin 6x dx 6 7n 1 7n 1 18

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Q: 1. (a) Prove the following. | (In x)"dx = x(In x)" – n (In x)-' dx

A: To prove that 1.a. ∫lnxndx=xlnxn-n∫lnxn-1dxb. Using 1.a compute the following      ∫lnx3dx

Q: Expand fix)= Si (x+ T),O<x< I ina half Yange sine series

A: We use the half range sine series formula to find the solution given as follows:

Q: Evaluate the integral. 7 Inx ey dy dx O 16 O 18 O 9 O 32

A:

Q: To prove that 2 dx 2 In V1 + (In x)² + In x| + C X/1+ (In x)² we use the substitutions

A: The given integral is ∫2 dxx1+lnx2 Find your answer below

Q: 16 9. Evaluate the integral 4 dx.

A:

Q: dy y(ln y- In a+1) dx 1 C = -In 1 C = In O Option 2 Option 1 1 C =-In None of the choices O Option 4

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Q: The integral 2N | 6xe- dx can be done with the substitution u = and du = dx.

A:

Q: Solve the folkowing di fferential equotions 1. xdy dx + dy x'

A: Firstly , we can solve for dydx. After that we can see that the equation is an homogeneous…

Q: Evaluate (а) S In(2.x + 2)dx -1

A: Given:          (a): ∫−11ln(2x+2)dx

Q: If In(ax)dx = 4, then a ( In(ax)dx %3D 1 4

A: we have to find the value of a

Q: 105. Prove 2* dx < 4 and 3-* dx VI

A:

Q: 1. Find b such that |(a) In 3. dx is equal to х (Б) 3.

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Q: Evaluate the double integral dy dx 4. 2.

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Q: Evaluate 1- x? dx 0. by Simpson's rale with n=5

A: Here fx=x1-x2 The interval is 0,1 and n=5. Hence the interval is divided into 5 equal parts.…

Q: (2) Compute dy dx if mxy=55

A:

Q: 1.) Evaluate In 2 e*2arctan e* A.) Jū+e2x) sec(2ªrctan e*) dx

A:

Q: (b) Evaluate In(x) dx 1

A: The given integral is divergent.

Q: 4. Prove that ах x" ea" dx xn- п-1 ‚ax dx а а

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Q: Integrate 2 In(11x) -dx . a) In (æ)? 2 11 + C b) None of these c) In (11æ)² + C 2 d) In (11æ)² + C…

A: topic - indefinite integral

Q: 8) The integral |4* dx is equal to 5x 5x 5x - +C C. In 5 А. -+C In 4 В. +C 5in4 D. +C 4 In5

A: Given ∫45x dxSubstitute u=5x → dudx=5→dx=15du=15∫4u duNow solving ∫4u duApply Exponential rule: ∫au…

Q: 5 + In x Evaluate dx. x(1 – In æ)

A: We need to evaluate the integral - I=∫5+ln3xx(1-lnx)dx We will solve it by substitution method.

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: