1- sin x a) Evaluate the limit by using L’Hopital's rule lim *x→7½ 1+ cos 2x b) Evaluate the integral by using the integration by parts rule [x sin xdx

Q: Use the limit definition of definite integral in the form of f(æ) dæ = lim {(#:)Ax, i=1 6 - a and r;…

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Q: Consider the function fcX) = Vx+8 Part A: us.ng a limit, s how thut f'l1)=- Part B: Find the…

A: Using definition of derivatives (limit form)

Q: 14 Find the limite by using L'Hopital Rule: X-I ①Lim 3x²-x²-2 sin 5x X 3 ③Lim 1-2 Sinx

A: Step 1: L'Hôpital's Rule: States that if the limit is in the indeterminate form of like…

Q: f(x,y)=(x+y) sin  1/(x-1) sin 1/(y+1) the limit of the function is 0 at the point (1, -1), the limit…

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Q: -1 -1 Differentiate xe 1 2 and hence integrate 2 dx.

A: Let g(x) =x e-12 x2 then x -derivative of g is given by dgdx=ddxx e-12 x2  now , using product rule…

Q: Compute the following integrals given that  ∫∞-∞ |f(x)|2dx =  ∫inf-inf|f^(s)|2ds: I)…

A: solution is given below:

Q: Express the limit as an integral. lim (8(x)²-3(x) ³)Ax over [0, 3] /=1 X ) dx

A: Express the limit as an integral.limn→∞∑i=1n(8(xi∗)2−3(xi∗)3)Δx over [0, 3]

Q: Why can't L'Hopitals Rule be applied to evaluate x - cos x lim x→+00 x + sin x

A: We have to explain about the given information

Q: Evaluate without use of Hopital's rule: tanx-sinx 1) lim-0

A: using expansion of trignometric function

Q: Use partial integration to derive that from image without deriving the right side. Here one can…

A: Let's prove, ∫xtan2xdx=xtanx−21​x2+ln∣cosx∣+C We can use the formula : ∫u dv=uv−∫v du Let's perform…

Q: How do we calculate the limit of limx→0  (sin(x))/(9x3), by using the Taylor approximation

A: limx→0  (sin(x))/(9x3)

Q: Evaluate the improper integrals or show that they diverge. Make careful use of limit notation. (a)…

A: Note:- Since you have asked multiple question As per our policy, we will solve the first question…

Q: sin z Prove that lim Cosz - = 1 and lim = 0. Z→0 Z Z→0 Z Hint: use the definition of the derivative,…

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Q: Use the reduction formula twice to find the value of this integral: integral tan^(5)xdx

A: We are entitled to solve only one questions, so I am providing you the solution of first question.…

Q: Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to…

A: We have to find the value of y(x) using frobenius method . The procedure is given below

Q: 7. lim vx² + x + 1 – vx² + 1 X00

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Q: Consider the weighted inner product defined by or functions f and g. (f,g) = [¹ f(x) g(x) dx, The…

A: To define the function inner_product(f, g), we can use the integral function to compute the…

Q: 4. Use the definition of integration to find (1+2x' ) dx , that is find lim f(x; )Ax

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Q: Evaluate the integrals sec 3x tan 3x dx

A: As per guidelines, i can answer only one question.

Q: x2 cos x+sin2x Find the limit: lim by means of the L' Hopital's |x sin x X-0 Rule.

A: We need to find the limit: limx→ 0x2cosx+sin2xxsinx. So,              limx→ 0x2cosx+sin2xxsinx=limx→…

Q: Use l’Hôpital’s rule to find the limits

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Q: 4) Compute the following limits: 2 b. limit Sin²x X-20 x LJ limit Sin'x C.J X 90 x²

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Q: The improper integral f(x)dx can be written as lim f(x)dx + lim_f," f(x)dx . ... W→ー0 w+00 If f(x) =…

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Q: (b) 3 – cos(x) 2

A: Follow the procedure given below.

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.