73. Use the definition of |x\, the chain rule, and the fact that (In x) = - for x > 0 to prove that (In |x) = - for all dx dx x+ 0.

Q: | (In x)" dx = x(In x)" – n[ (In x)"' dx n-1

A: Use integration by parts

Q: Q2: - If y = 3sec3t and x = etan3t prove that: - dy (In3)2 + dx y * logy x *

A:

Q: 10. Evaluate the integral. dx x²/4 – x²

A:

Q: 4. Evaluate the improper integral dx +e"

A: We know that, ddxex=ex ∫1a2+x2dx=1atan-1xa+c tan-1(1)=π4 tan-1(∞)=π2

Q: 2- Find O for: y= In (x+5x) dx

A: Tpoic- Differentiation

Q: 3. Use the Comparison Test to determine whether the integral cos? x dx is convergent or divergent.…

A:

Q: what is the derrivative of y^3 + y^2cos(x) = x^3 -3x and what points in the function for dy/dx does…

A:

Q: Find the indefinite integral.  (7-e^x)/e^3x dx

A:

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

A:

Q: Demonstrate that for the integral of p(x)/q(x) where p(x) is of degree 2 and q(x) of degree 3 it is…

A:

Q: 1. If f(x) dx = 3, evaluate [[1 + 21(x)] dx using the properties of interga 2. If f(x) dx = -2 and…

A: As asked we can answer question no 1 and 4 as below.

Q: Derive a formula for In x dx where a. nt-1 b. n=-1

A:

Q: - Evaluate the integral dx using Simpson's th rule e* – 1 3 -

A:

Q: 4х+2 Evaluate the integral -dx. (Hint: Use Algebra to help further rewrite your (x² +1)(x=1)…

A: First break by using partial fraction then integrate

Q: 9 Evaluate the integral f 2x (x² + 1)° dx using the substitution = x² + 1. и — с 2.x (x? + 1)° dx +C

A:

Q: 2. * S arctan x dx 1 In(x²) + C C 1. x arctan x C A x arctan x -In(x² + 1) + C -- B 1 D x arctan x…

A: Solve the given integration

Q: Evaluate dx V In(8-x)) at x = 1. = 1.

A: We have to find: ddxln(8-x2)3   at x = 1.

Q: Q2: - If y = 3sec3t and x = etan3t prove that: - dy (In3)² dx y logy x * * -

A:

Q: 2 Evaluate the improper integral e2a dx, if exists.

A: Evaluate the improper integral  ∫-∞2 e2x dx

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: Show /x^ In х dx xn+1 ((п + 1)In x — 1) (п+1)2

A:

Q: 60. Is there a value of b that will make Sx + b, x<0 (cos x, x20 continuous at x= 0? Differentiable…

A: A function is continuous at x=c, if LHL at c=f(c)=RHL at climx→c-f(x)=f(c)=limx→c+f(x) A function is…

Q: Q2: - If y = 3sec3t and x = etan3t prove that: - dy (In3)² dx y * logy x |

A:

Q: (3) Evaluate the definite integral e + e -dx

A:

Q: d41 -cosh(x) at x : O is dx41

A: cosh (x)

Q: 1. Evaluate the indefinite integral. Hint: U-substitution. 3 dx x In x

A: ∫ 3x lnx dx 3∫ 1lnx·1x dx

Q: if p> 1, (diverges; if p <1. 0o dx Prove that f p- 1 хР 1 Hence evaluate: dx. x6

A:

Q: 6. Evaluate the integral. x³e2x dx

A:

Q: 27. For what value or values of a does ах |dx 2x x² + 1 converge? Evaluate the corresponding…

A:

Q: find (In x)³ dx 2x In (In x) dx x In x 11. 12. In9 et/2 dx tan x In (cos x) dx 13. 14. In4 eVi cdr…

A:

Q: In 3-Jın Evaluate the following: dx 2e*cose*dx T/3 4 sinx dx 1-4COSX 2-Jo x(lnx)² 1- Jo" x(Inx)?…

A: Given that three different types integration we solve these integration by using general formulas of…

Q: 1. Evaluate the integral: x/x – 1 dx.

A: answer = 1.0667   explanation is given in below step

Q: Sx". .n-1 + n[x²- x" sin x dx = -x cos x + n cos x dx

A:

Q: Evaluate the definite integral. e2 - x dx

A: The given integral is as follows. Let u=2-x. Then, du=-dx. The Lower limit will be u=2-2=0 and the…

Q: 2. Evaluate dy/dx for y= x In x 1+inx

A:

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

A: The given integral is divergent.

Q: In Evaluate the integral: Sé e x²+3x-7 x-2 dx

A: Option (b) x22+5x+3ln|x-2|+C

Q: ormulas to compute the following antiderivative. x-1 x² -dx.

A:

Q: 5. prove that d (-x) (+ x) = X dx dx d² dx² - x² +1

A:

Q: 169 dx Evaluate the integral x( In x)° 13

A:

Q: 12) a) Use either the Trapezoid OR Midpoint Rule to estimate [ In(1+e* ) dx using at least four…

A: “Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: If In |(In x)" dx, show that In = x(In x)" – n.Ip-1- Hence find nx)* dx.

A: I am going to prove the given identity by using integration by parts and then use this identity to…

Q: 1. Evaluate the integrals that converge: dx 2x ! dx (x2 + 5)3 b) a) 3-x - 00

A:

Q: - if eV = In(x) + In(y) + e³ ху 3 Find dx

A:

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.

Q: 1. Integrate by parts and evaluate x cOS x dx 0, а. b. In(x²) dx 1 In x dx x5 С.

A: By using Integration by part  ∫udv=uv−∫vdu

Q: 1. Use algebraic manipulation and (if necessary) u-substitu- tion to integrate the function. (a) /…

A: Hey, since there are multiple questions posted, we will answer the first three subparts of the…

Q: 5. Find the integral | 2(x* - x°) (x5 – 1)®dx x In(x + 2) dx а. d. | In x? dx | 2xe** dx b. е. 1 2x…

A:

Q: Q2: - If y = 3sec3t and x = etan3t prove that: - dy (In3)? dx y logy x

A: