** The definite integral " - 1/4 • 1/4 tan-1x² + /x²+ 1 Cosx +2 dx is equal to A 1 • T1/4 tan-1x²+/x² + 1 dx B Cosx +2 1/4 tan-x² +V x< + 1 CoSx + 2 D none of these

Q: Integration techniques: Use whatever method you like to solveeach of the given definite and…

A:

Q: 13. To evaluate f 3cos*x-1 dx. the integrand can be transformed to 1-sin2x a. [(3cos x+ csc²x)dx b.…

A:

Q: 1. Interval, 0 < x < 2; function, f(x) 0 < x < 1, 1 < x < 2. = 2 – x, 4 1 ANS. f(x) 1 +* E 2 cos – 1…

A: Fourier Cosine series expansion of the function

Q: Evaluate the integral (sin (x/7) dx .2 49-x2 2 sin -1(x/7) dx /49-x² %3D

A: ∫sin-1x/7249-x2dx

Q: 1. f(x) = sin 2x + cos 3x + x² = 0 with xo = -0.5 , Ax 0.1, TOL = 0.001, in [-0.5,1] %3D

A: Given equation is fx=0 where f(x) is defined by fx=sin2x+cos3x+x2  Given initial solution is x0=-0.5…

Q: a 1 - x²) cos(nx) dx F(x 一九 4 sin(nл) — 4nл соs(nл) — 2n²л² sin(n) πη3

A:

Q: Increasing and decreasing functions Find the intervals onwhich f is increasing and decreasing.

A: We'll answer the first question only since the exact one wasn't specified. Please submit a new…

Q: Evaluate the integral: 25 tan² 12x dx Choose the most appropriate answer. tan3 3 12x O 25 × 1 12 ln…

A: Explanation of the answer is as follows

Q: 7. Integrate: S sec? (n²x)dx A. 2n tan n² + C В. x sec2 п2 + С C.금tan(T2x) + C D. x sec nx + C 8.…

A:

Q: Ruestion 13 Evaluate the integral. x4 - 3 dx 2 16 + x o 3.253x + tan 1주 + 253 글x-16x + tan1 +C 16x +…

A:

Q: 8. f(x) = 2sin² 3x a. 12cos(3x)sin(3x) b. -12cos(3x)sin(3x) c. 12sin(9x)cos(3x) 9. f(x) = 3 cos² a.…

A:

Q: In( cosh(2x) + sinh(2x) ) + In( cosh(2x) - sinh(2x) ) = Select one: O zero O 2x O 2 O None O- 2x O 2…

A: Topic- hyperbolic function and log

Q: 2 sin -1(x/ 4)dx Evaluate the integral ! V 2 16 -x ( sin -(x/4) dx V16-x 1 2

A:

Q: Evaluate f x² cos 4x dx Evaluateſ sin* 3x cos3 3x dx Evaluate the integral J x²+x-1 dx x(x²-1)

A: Given:   i) ∫x2cos 4x dx      integrate by parts ∫fg'=fg-∫f'g     here f=x2,f'=2x , g'=cos…

Q: Steb by step

A: We will need integration by parts for this problem. So we assume u and dv and from there find du and…

Q: 1. n! n=0 2. (-1)"x2n (2n)! 3. (-1)"x2n+1 2n + 1 n=0 x2n+1 4. E(-1)"- (2n + 1)! n=0 A. sin(x) B. e*…

A:

Q: Using integration by parts, we can convert the integral x200 . sin x dx to which of the following?…

A:

Q: hich the extrema occur. g(x)=x^2+cos⁡(x^2 ) on [-√π,√π]

A: The function given is gx=x2+cosx2 on the interval [-√π,√π]. To find the extrema of the function…

Q: the integral expression on the left to an equal expression on the right. (cos(x)-cos(7x))dx ots…

A: We will solve the integral.

Q: sin In 4+tan cos-dx 4+tan-Icos- 2 1+ cos 2sin In4 4+ tan cos- dx A 1+cos 4+ tan-cos- 2sin-In 4+…

A: To rewrite the given integral so that the integral can be evaluated easily. ∫sin…

Q: Find the indefinite integral.

A: Given,∫2sinxcosxdxwe need to find the indefinite integral

Q: II. Definite Integral x²dx 1. 3 (1-x3)7 2. sin³ ydy

A:

Q: 3. Find the critical numbers of the function f(x) = sin2 3x + cos 3x , 0 <x < 2/3:

A:

Q: (1) (sinz) + (cosr)* sin x COS x (2) tan x in terms of sin x, cos x %3D d. d. ( sin x COS X %3D dx…

A:

Q: 9. The value of f 1-x 1+x A. sin ¹¹x + √1 + B. sin¹x - √1 + dx x² + C x² + C C. sin ¹¹x + √1- x² + C…

A:

Q: CoS cos³ 0 5./ d0 1-sin0

A: Given problem:- ∫cos3(θ)/(1-sin(θ)) dθ

Q: The integral of the given function csc x dx is equal to 3cos' x-10sinx cos'x-15sin xcOsx 15sin'x…

A: To evaluate the given integral.

Q: 1- cos 2x 2 1+ cos 2x 1 (1- cos? 2x)(1+cos 2x) dx = 8. 8.3.25 sin2 z cos a dx = 1+ 8 dx = cos 2x -…

A:

Q: (1) y = In(x+ 4)- x tan (2) xy + xy x + 3y (3) Iny e' sin

A: In this question we will use basic of calculus i.e concept of differentiation. We will simply…

Q: 1. Integral of x sin^3 (x^2) 2. Integral of sin^4 3x

A:

Q: Find the indefinite integral ∫ 2sin x cos x dx

A: The given integral is ∫2sin(x)cos(x)dx

Q: Use the Intermediate Value Theorem to show that 2x – 3 - if – 3 0 has a zero on the interval [-2,…

A: First compute the function if they are continuous then apply the intermediate value theorem.

Q: 3 In(1+ V sin²x .limr→0 V1– cos x

A: Consider the given expression. limx→0ln1+sin2x31−cosx Put the limit in the expression.…

Q: 1) / sen(x) 4+ cos²(x) I 1+4x² 2) [√5/2 S² 3) 0 4) 1 x² - 2x + 2 dx √ √2/2 are 0 X dr √2/2 arccos(x)…

A:

Q: я/4 4 sin (4t) dt. 9- cos (4t) Evaluate the integral я/4 4 sin (4t) dt%3D 9- cos (4t)

A: Use integration by parts

Q: (10) Find the indefinite integral .csc². -dt cot x sec x tan x sec x -1 1 (S- dx (1+ x' 1/3 + 2e* +1…

A:

Q: 1. f(x) = 2x² – 3x + 4 2. f(x) = e-2x sin 5x 3. f(x) = x cos7 x 4. f(x) = 5 sin 3x – 17e-2x cos 3x…

A:

Q: 2. x²+x+2₂ x²-1 -dx

A: Follow the procedure given below.

Q: 12) f)=sin(x) 0<x<1 Ans. 24 1 (₁ A 7 (1)(3) 1 -cos(2x) +- -cos(4x) (3)(5)

A:

Q: 1. Evaluate the indefinite integral cos x sin³ xdx. 2. a) Evaluate the indefinite integral cos“ xdx.

A:

Q: cos(9x) -dx COS J 1+ sin²(9x)

A:

Q: 8. +1 (x-1)³ -dx

A: Integration

Q: 38. f(x) = 3(sin-1 x)² %3D 40. f(x) = tan1(Inx) %3D

A:

Q: Match the integral expression on the left to an equal expression on the right. cos(7x))dx + cos(…

A:

Q: derrivative of: Cos(xy)+secy=5

A: Consider the given expression as cosxy+secy=5

Q: (x + 1)³sinx dx x+1)° cosx+3 (x + 1)2 sin x-6 (x + 1)cosx-6 sinx+ C | (x + 1) cos x 3 (x + 1)2 sin x…

A: Solving the problem by finding the integral by oarts method.

Q: In2| 4+ tan-'cos- dx 2 sin ( 1+cos | 4+ tan-'cos- 2sin- · In²| 4+ tan-'cos- dx 2 (A 1+cos- 4+…

A:

Q: Identify the correct form of the indefinite integral tan (x) csc (x) dx sec-/(x) +C O tan-(x) + C…

A:

Q: cos² x d.x 4 sin* x 2.

A:

Q: Evaluate using Integration by Parts. cos(13x) dx (Express numbers in exact form. Use symbolic…

A:

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.