Ex 15: Prove that : d du a) tanh u = sech² u dx dx

Q: S (x, y) = sen (r² – 0²) + In(r® + 0), con Vr? + y? A r = = arctan

A: Given that, f(x, y)=sen2r2-θ2+lnrθ+θfor r=x2+y2and θ=arctanyx

Q: Tan Be Xe O de d Bo find

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Q: Ex 15: Prove that : d du tanh u = sech2 u dx dx Proof:

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Q: Ex:57 Find the FisiE of the Juuctiou sin TIt ZL oくtくし - siu -Lくtく

A: Given, The formula for the Fourier series expansion is,

Q: If z = f(x, y), where x = r cos 0 and y = r sin 0, show that a²z 1 a°z 1 əz r дr дх? dy? дr?

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Q: Find the area outside "r^2=9cos(2theta)" and inside "r=2-cos(theta)"

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Q: The suitable substitution to evaluate the integral /tan°(2x) dx is cos“(2x) O u= tan(2x) O u=…

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Q: Sketch and evaluate the reverse in tegral 9/x dy dz

A: Consider the provided integral,

Q: If z = cos (r + cos y), show that az Fz ar ərðy dy dr

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Q: Prove that J seccu) du - Ln/Sec (u)+ tan (w) + C

A: Integrated in general

Q: if y=tan x,prove that y'''=2(1+y²)(3+y²)

A:

Q: Ex: 1) Let f(x, y) = sin(xy) X to wake fa cont. function on all of R²? ,x = 0. How should you define…

A: The given function is fx,y=sinxyx, x≠0 and f(x,y) is continuous on ℝ2. To find: f(0,y).

Q: O! Find tue indefinite integrel by q meleing tue febstelion d=4 Sin o. 2 *7 16-x"

A: To determine: by substituting x = 4sinθ

Q: ind the length of the curve x cos(t) + In(tan(;t)), y = sin(t), /4<t< 3n/4. %3D ength =

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Q: make the given substitutions to evaluate the indefiniteintegrals. ∫sec 2t tan 2t dt, u = 2t

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Q: let Po)- tar'c) + tant (L). show that Plx)=C,where cis a constant and then Aind C

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Q: L[cosh(@ot) u(t)], C[sinh(@et) u(t)] and L[cos (@t) u(t)

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Q: as Let ĝ +sinh (z?) - ji = 27 where w=f (x,y,2)- %3D Find 2 nd JW and au

A: In the given question we have to find the partial derivatives of with respect to x ,y and z. here we…

Q: dy If tan(ry) + 5 = ry then show that dr? 2y

A: Differentiation two times

Q: π/3 So, f²ef/³ t In u tans ds du dt

A: Given problem:-

Q: dz and du dz Find when u = In 2, v = 1, if z = 10 tan'x, and x= e" + In v. -

A: Here we the chain rule to get the answer .

Q: Calculate the area of the circle r = 3 sin θ bounded by the rays θ = π/3 and θ = 2π/3.

A:

Q: Find the area inside both curves. r = 2+ sin(20),r 2+ cos (20)

A: If f is continuous and non negative on the interval α≤θ≤β with 0&lt;β−α≤2π. Then area of the region…

Q: d - [sin(t) dt = 0, Where k is cons tan t dx Select one: True False x

A: Here, the integral has the constant limits 0 and k. So, ∫0ksin(t)dt s in terms of k that is…

Q: If u = tan- the value of x2 +y? +xy du du at x = 1 and y = 3 is

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Q: cos ve J Vē sin? ve de = 2 sin ve 1 jus O 2 + c sin ve

A: Given ∫(cos(√θ)/√θsin2√θ)dθ   Let √θ= t     Integrate on both sides     (1/2√θ)dθ= dt            dθ=…

Q: Te 2=tnx then prove thot dx 28 Snメー tty? 1+xて

A: We have to use proper calculus applications.

Q: If y = tan x, prove that y``= 2(1+y² )(3+ y²).

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Q: Prove that ∫ cot u du = ln∣sin u∣ + C.

A: Prove that ∫ cot u du = ln∣sin u∣ + C

Q: 10. Prove the following formulas : d cot u = – csc u. dx d CSC и 3 — csc и.cot и. dx 2 du а ) * dx…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Find in terms of x and y if arctan(r + x²y) = ry

A: Given that         arctan(x+x2y) = xy We have to find dy/dx in terms of x and y.

Q: r cot VE-tan° V cosº /T VEsin JE dx

A: Given that: ∫cot3x-tan6x.cos6xx.sin3xdx

Q: the points on aut puit +y" + z? =I that and farthest from the po int (R,!,2) the are close ot to 2.

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Q: Let P be outside C(O,r) and S, T e C(0,r) so that PS and PT are tangent to C(0, r). Show that PS =…

A: Given that PS and PT are tangents. We know that the tangent to a circle makes a right angle with the…

Q: Find the area enclosed by one "leaf" of the three-leaved rose: Bepaal die area wat deur 'n "blaar"…

A: Find the area enclosed by one leaf of r=4cos15θ such that -π30≤θ≤π30

Q: If y= tan x , prove that y`= 2(1+ y² )(3+ y²).

A:

Q: The suitable substitution to evaluate the integral tan (2x) dx is cos“(2x) none O u= csc(2x) O u=…

A: Given: ∫tan52xcos42xdx

Q: Verify the formulas d u' -[arcsin(u)]: dx Vi-u d u' -[arccos(u)]=- dx V1-u? d -[arc cot(u)]= dx u'…

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Q: EXpress ume trig fatIOS as TTACCTIOTIS TII Smmpiest teTIIS. 16 R. V33 17 sin R = Cos S sin R and cos…

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Q: 1. Use Trigonometric substitution to evaluate f dx. Hint: Ju? – a² let u a sece.

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Q: 10. Prove the following formulas: du cot u %3D-сsc и. 2. а ) a dx d dx du csc u = - csc u.cot u. b)…

A: (a) ddxcot u=dducos usin u·dudx=dducos u×sin u-cos u×ddusin usin2 u·dudx=-sin u×sin u-cos u×cos…

Q: D. If z = In(x – y) + tan(x + y), show that Z = Z y XX

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Q: Prove the following hyperbolic trig identity. cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y).

A: cosh(x+y)=cosh(x)cosh(y)+sinh(x)sinh(y)

Q: 213 If cot 0+ tan = and sece - sece = y, Prove that (x'y)° (xy²)*° = %3D

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Q: use the definition, prove that d sin z = cOs z dz

A: Topic = Derivative

Q: If y= tan x, prove that y 2(y² +1)(3y² +1).

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

Q: 1. If y= tan x, prove that y= 2(y´ +1)(3y² +1).

A: Use the following differentiations as, ddx(tanx)=sec2(x)ddx(sec(x))=sec(x)tanx Now, differentiating…

Q: Use the substitution u = tan x to evaluate dx 1+ sin? x Hint: Show that dx du 1+ sin? x 1+ 2u2

A: Given that: The integral is ∫ dx1+sin2x

Q: Evaluate F· dr C where F(х, у) %3D хуі + уј C:r(t) = 4 costi+ 4 sin t and 0 <t<.

A: Given that , Fx,y=xy , yrt=4cost , 4sint and 0≤t≤π2 Evaluate ∫CF.dr=?

Q: Consider that sin(x) = e, where %3D 0<x< T, then dy tan (x) dx False O TrueO

A:

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.