Question 4. Evaluate the following: (a) /æ sin æ dæ, (b) / tan (2) dæ, 2 - Væ (c) lim 2-4 x2 – 16 (d) / 10/n="d:

Q: y"+y' - 2y = x2

A: Solution :-

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A: Given ∫ (a2- x2)32 dx Perform trignometric substitution x= aSin (μ) →υ= arcsin (xa), dx= acos (μ)…

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A:

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A:

Q: S(1+secx)³dx

A:

Q: 5.[(2√x -3x)² dx

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Q: e* (x + 2) (x+3)² dx

A: To evaluate the following integral.∫exx+2x+32 dx

Q: x +4 x² + 2x + 5 -dx

A: To calculate the integral ∫x+4x2+2x+5dx.   Use the common integrals:…

Q: dy If y = (5x + 1)(2x +In x), then will be %3D dx (5x + 1) +) + 5(2x+ In x) A. -1 B. (5x+ 1) - 2. +…

A: The given problem is to find the derivative of given function y=(5x+1)(2x-1+ln(x)) , we can use…

Q: d, 3 Find dx -)- + x(х - 1)). A -3 +2х-1 B +1 -3 +1 -6 +2х-1 x3 (D

A:

Q: 3 2.)f x³ (4 + x²)² dx ; Use rationalizing substitution

A:

Q: 2-fx (sec²x)dx

A: Given:

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A:

Q: 2х -3 -dx. Evaluate(x-2)(x² + 1)

A:

Q: 1-x dx (x- 3Xx-2)

A:

Q: 9. Simplify sec² x - 1 sec² x

A: secx=1cosxI=sec2x-1sec2xsin2x+cos2x=1

Q: dx 4 (7-x²)*

A: We will find the integral as following.

Q: 1. (a) Determine + 2x2 dx (b) Determine + 2x2 dx

A:

Q: |(5z* + 3z*) (4x³ + 3x*)° dæ

A:

Q: S sec²(3 – 2x)dx

A: ∫sec23-2x dx Put, 3-2x=t⇒0-21=dtdx⇒-2 dx=dt⇒dx=-dt2

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A:

Q: 9-x2 dx x2 Evaluate:

A:

Q: 2. secx d X secx

A: ∫sec2x5-sec2xdxPrepare for substitution, use:sec2x=tan2x+1

Q: хt — х +6 dx х3 + Зх

A: We shall use polynomial long division method to simplify our integrand.

Q: Use Use your Exponent Rules to Simplify (4r* y" 2² ) (xy") 4a° y1 3 4.x 213 ,10 46,10 6,10

A:

Q: Evaluate 4 fa²¹ t tan³ (x5) dx

A:

Q: A jungle has 400 monkeys, and it is estimated that the maximum population of the monkeys in the…

A:

Q: x (1+x²) dx

A: Let, I=∫x1+x2dx  ............(1) Substitute, u=x2+1 dudx=2x dx=12xdu Put these values in equation…

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A:

Q: Q2\ Evaluate : fot√x dx

A: The given integral is ∫cot2xxdx use the identity  cosec2x-cot2x=1 cosec2x-1=cot2x therefore the…

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Q: coth (7x - 3) + C B) tanh (7x - 3) +c I sech? (7x - 3) + C tanh (7x - 3) + C

A:

Q: * Drive the following function for :(y) (5xy²z)²/³ * (sinxz) 20(sinxz) (xyz) 3(5xy²z)¹/3 A 10(sinxz)…

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.