eCoS ecos x sin x dx In x 2. Vx 3. tan'x dx

Q: 5. S sec(3x) tan(3x) dx

A: formula:∫ secx tanx dx =sec x +Cwe are given;I=∫ sec(3x) tan(3x) dx3x =t3 dx =dtdx =dt3

Q: integral: 5x sec2 (x) dx

A: ∫ 5 x sec2xdx

Q: sin (In x dx ecotx

A: Given- ∫sinlnxecotxdx

Q: 1 dx 1-sinx

A: We will find the integral as following.

Q: dx sinh x - cosh x

A: To Evaluate: ∫dxsinhx-coshx Concept: i. ∫afxdx=a∫fxdxii. ∫exdx=ex+C

Q: sech (In x*) 15. dx x/6 tanh (In x2) – tanh? (In x2)

A:

Q: (tan x) = fu. v dx = (sin x) = | -=-=dx = (cos x) = fe" dx = | (arcsinx) = sec x tan x dx = secx tan…

A: We have to find out derivative and integral

Q: 2)/ sinh cax )x

A:

Q: ftan (x}sec"(x)dx Scot (x)csc°(x)dx Ssin (x)cos°(x)dx

A:

Q: + sin(xx) – 2er )dx + sin(x) – 2e* )dx

A:

Q: Prove: sinh(x – y) = sinh(x) cosh(y) – cosh(x) sinh(y)

A: The basic relations we need to know to solve this problem are, sinh(x) = ex- e-x2cosh(x) = ex+ e-x2

Q: integral tan3(x)sex(x)dx

A:

Q: ∫(2 cos(x) − 7sinx(x))dx   Evaluate indefinite anti-differentiation

A:

Q: sec (In(x) 1. dx 2. Se tan(2x) sec²(2x)dx

A:

Q: 15. sinh 2x = 2 sinh x cosh x %3D

A:

Q: (sinh x + cosh х)*3 еx

A: Solution:-

Q: (Integral sign) tan^2 x cos^3 x dx

A: The given integral is

Q: sinh x + cosh x)ª= e4x

A:

Q: 11) (y)* = (In tanx)(x*)(sin x)* %3D

A:

Q: 11) (/y)* = (In tanx)(x*)(sin x)* %3D

A:

Q: sin(In x) dx

A:

Q: cosh x sinh x dx 2 (cosh x + 3)

A:

Q: x' sin (tanx*) -1 Integrate 1+x

A: Introduction :

Q: y = tan In(x + sin x) %3D y = sin x + In sin e* %3D

A: y=tan ln(x+sinx) y=sinx+ln sinex

Q: (2) y = /sinh /tan x

A:

Q: (8 sin(x) + 7 sinh(x)) dx

A:

Q: integrate (sin(x)*tan(x)-cos(x))/tan (x+1)

A:

Q: e-V2x sinh x sin x x dx

A:

Q: S tanx 1+ COS X JX

A: Given problem:-  ∫tan(x)dx/(1+cos4(x))

Q: sin 2x+sin x dx Vsin2x -cos x v.

A:

Q: cos*(x)dx

A: According to integration reduction formula ∫cosn(x) dx=n-1n∫cosn-2(x) dx+cosn-1(x) sin (x)n . For…

Q: Show that arctan(tanh(x)) = sech(2x). arctan(tanh(x)) dx D²tanh(x)) %3D 1 + ])-: 1/cosh²(x) 1 +…

A:

Q: y = Arctan (sinh x)

A: dydx=ddx(arctan(sinh x))

Q: Integral of dx/sinx+tanx

A:

Q: dx sinh x + 2 cosh x

A:

Q: dy sec x.tan х dx 1- sec' x

A:

Q: 2 *sinh“ x – sinh x+ cosh x dr -dx cosh¯ x

A:

Q: 3 sinx+cos x Evaluate S dx. 2 sinxcos x

A:

Q: valuate f sin³ 2x cos³ 2x dx.

A:

Q: sin(nx)dx |sin(nx)dx (xcos(nx)dx

A:

Q: (cos(2x) – sin(2x) cos(2x)) cos(x) – sin(x) For -dx%3D +C. What is F (x)?

A:

Q: sin x dy (1 + y) cos x + sin x If y=- , prove dx %3D cOS x 1+2y + cos x-sin x 1+ sin x 1+ 1+ cosx 8.…

A:

Q: (3 sin x - sin 3x) dx

A: To integrate: ∫(3sin(x)-sin(3x))dx Solution: We know,∫sin(ax)dx=-1acos(ax)+C Using above standard…

Q: Y= Gas"(Sin-(tan

A: Given that: y=cos-1(sin-1(tan-1(csc-1x)))

Q: o( san(s) sin(2x) + lim X 0 cos (x) × 6

A:

Q: a) / sin°(x) cos(x) dx = » / sin(x)(cos(x))² dx =

A:

Q: 1 (sinx\1-cosx lim X→0

A: We need to evaluate limx→0sin xx11-cos x Let y = sin xx11-cos x Then log y = 11-cos x log sin x…

Q: integral sin -1 x dx

A: Given Data The integral is ꭍsin-1xdx.   Let,

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.