
MathHello! Probably we need to find all the terms of this progression. Recall that each next term of a geometric progression is obtained by multiplying the previous term by the quotient, denote it as...

MathThis question requires you to set up a system of equations. First, you must identify your variables. Since we want to know how many adult tickets and how many child tickets were bought, those...

MathHello! Actually, the function `x` (absolute value of `x` ) is defined as a piecewise function: `x = {(x if xgt=0),(x if xlt0):}` There are two "pieces" on which this function is defined using...

MathAn expression represents a rational number if the simplified result yields a number of the form`p/q` where p and q are integers and `q!=0` A square root is rational only if the expression in the...

Math(1) The curve created by a chain hung from fixed points is called a catenary which is the graph of the hyperbolic cosine. The St. Louis Arch is based on an inverted catenary. Catenaries were...

MathHello! I suppose that the age of each child is a natural (whole) number. Then from (b) we get that Alejandra's age is a natural number greater than 8 but less than 10. There is the only such...

MathGiven `dy/dx = 1/sqrt(4x^2) ` and `y(0) = pi` We have to find y. So we can write, `dy=dx/sqrt(4x^2)` Integrating both sides we have, `y=int dx/sqrt{4x^2)+C` Now let `x=2sint ` So, `dx=2cost...

MathRecall that` int f(x) dx = F(x) +C` where: f(x) as the integrand function F(x) as the antiderivative of f(x) C as the constant of integration.. For the given problem, the integral: `int...

MathComplete the square at the denominator: `x^4 + 2x^2 + 2 = (x^2)^2 + 2x^2 + 1 + 1 = (x^2 + 1)^2 + 1.` Now we see the substitution `y = x^2 + 1,` then `dy = 2x dx,` and the integral becomes `int...

MathWe have to evaluate the integral: `\int \frac{1}{(x1)\sqrt{x^22x}}dx` We can write the integral as: `\int \frac{1}{(x1)\sqrt{x^22x}}dx=\int \frac{1}{(x1)\sqrt{(x1)^21}}dx` Let `x1=t` So...

MathRecall that `int_a^b f(x) dx = F(x)_a^b` : `f(x)` as the integrand function `F(x) ` as the antiderivative of `f(x)` "a" as the lower boundary value of x "b" as the upper boundary value of x To...

MathTo evaluate the given integral:` int 2/sqrt(x^2+4x)dx` , we may apply the basic integration property: `int c*f(x)dx= c int f(x)dx` . The integral becomes: `2 int dx/sqrt(x^2+4x)` We complete the...

MathBy completing the square and making simple substitution, we will reduce this integral to a table one. `x^24x = (x^2 + 4x + 4) + 4 = (x+2)^2 + 4 = 4  (x+2)^2.` Now make a substitution `y =...

MathWe have to evaluate the integral:`` `\int \frac{2x5}{x^2+2x+2}dx` We can write the integral as: `\int \frac{2x5}{x^2+2x+2}dx=\int\frac{2x5}{(x+1)^2+1}dx` Let `x+1=t` So, `dx=dt` Now we can write...

MathFor the given integral: `int 2x/(x^2+6x+13) dx` , we may apply the basic integration property: `int c*f(x) dx = c int f(x) dx` . `int 2x/(x^2+6x+13) dx =2 int x/(x^2+6x+13) dx` To be able to...

MathTo evaluate the given integral: `int_(2)^(2)(dx)/(x^2+4x+13)` , we follow the first fundamental theorem of calculus: If f is continuous on closed interval [a,b], we follow: ` int_a^bf(x)dx =...

MathTo be able to evaluate the given integral:` int_0^2 (dx)/(x^22x+2)` , we complete the square of the expression:`x^22x+2` . To complete the square, we add and subtract `(b/(2a))^2` . The...

MathRecall that `(arccos(x))' = 1/sqrt(1x^2)` and make the substitution `y = arccos(x),` then `dy = 1/sqrt(1x^2).` The limits of integration are from `arccos(0) = pi/2` to `arccos(1/sqrt(2)) =...

MathWe have to evaluate the definite integral: `\int_{0}^{1/\sqrt{2}}\frac{arc sinx}{\sqrt{1x^2}}dx` Let `t= arcsinx` Differentiating both sides we get, `\frac{1}{\sqrt{1x^2}}dx=dt`...

MathMake the substitution `u = sin(x),` then `du = cos(x) dx.` The integration limits for `u` are from `sin(0) = 0` to `sin(pi/2) = 1,` and the integral becomes `int_0^1 (du)/(1 + u^2) = arctan(1) ...

MathWe have to evaluate the integral: `\int_{\pi/2}^{\pi}\frac{sinx}{1+cos^2x}dx` Let `cosx=u` So, `sinx dx=du` When `x=\pi/2, u=0` `x=\pi, u=1` So we have,...

MathMake the substitution `y = e^(x),` then `dy = e^(x) dx` and `e^(2x) = y^2.` The limits of integrations for `y` become from `e^(ln2) = 1/e^(ln2) = 1/2` to `e^(ln4) = 1/e^(ln4) = 1/4.` The...

MathFor the given integral problem:` int_0^(ln(5))e^x/(1+e^(2x))dx` , it resembles the basic integration formula for inverse tangent: `int_a^b (du)/(u^2+c^2) = (1/c)arctan(u/c) _a^b` where we let:...

MathMake the substitution `u = sqrt(16x^2  5),` then `16x^2 = u^2 + 5,` `du = (32x)/(2sqrt(16x^2  5)) dx = (16 x dx)/sqrt(16x^2  5),` or `dx/sqrt(16x^2  5) = (du)/(16 x).` The limits of...

MathTo be able to solve for definite integral, we follow the first fundamental theorem of calculus: `int_a^b f(x) dx = F(x) +C` such that f is continuous and F is the antiderivative of f in a closed...

MathMake the substitution `u = sqrt(4x^2  9),` then `du = (4x)/sqrt(4x^2  9) dx.` Inversely, `dx =sqrt(4x^2  9)/(4x) du = u/(4x) du` and `4x^2 = u^2 + 9.` The limits of integration become from...

MathMake a substitution `y = 2x,` then `dy = 2 dx` and the limits of integration for `y` are from `0` to `3.` The integral becomes the table one: `int_0^3 1/(1+y^2) (dy)/2 = 1/2 (tan)^(1)(y) _(y=0)^3...

MathWe have to evaluate the integral : `\int_{0}^{\sqrt{2}}\frac{dx}{\sqrt{4x^2}}` let `x=2sin t` So, `dx=2cos t dt` When x=0, t=0 and when `x=\sqrt{2}, t=\pi/4` So we have,...

MathMake a substitution `x = 3y,` then `dx = 3 dy` and the limits of integration for y are from 0 to 1/18. The integral becomes a table one: `int_0^(1/18) (9 dy)/(sqrt(99y^2)) = int_0^(1/18) (3...

MathWe have to evaluate the integral:`\int \frac{x2}{(x+1)^2+4}dx` Let `x+1=u` So, `dx=du` Hence we have, `\int \frac{x2}{(x+1)^2+4}dx=\int \frac{u3}{u^2+4}du` `=\int...

MathWe have to evaluate the integral `\int \frac{x+5}{\sqrt{9(x3)^2}}dx` Let `x3=u` So, `dx=du` Hence we can write, `\int \frac{x+5}{\sqrt{9(x3)^2}}dx=\int \frac{u+8}{\sqrt{9u^2}}du`...

MathRecall indefinite integral follows `int f(x) dx = F(x)+C` where: `f(x)` as the integrand `F(x)` as the antiderivative of `f(x)` `C` as the constant of integration. The given problem: `int...

MathThis integral may be taken by dividing into two parts: `int (x dx)/(x^2 + 1)  3 int (dx)/(x^2 + 1).` The second integral is `3arctan(x) + C.` The first integral requires the substitution `u =...

MathFor the given integral: `int 3/(2sqrt(x)(1+x)) dx` , we may apply the basic integration property: `int c*f(x) dx = c int f(x) dx` . `int 3/(2sqrt(x)(1+x)) dx = 3/2int 1/(sqrt(x)(1+x)) dx` . For...

MathRecall that `int f(x) dx = F(x) +C ` where: f(x) as the integrand function F(x) as the antiderivative of f(x) C as the constant of integration.. For the given problem, the integral: `int...

MathWe have to evaluate the integral: `\int\frac{sinx}{7+cos^2x}dx` Let `cos x=u` So, `sinxdx=du` Hence we have, `\int \frac{sinx}{7+cos^2x}dx=\int\frac{du}{u^2+7}` `=\int...

MathWe have to evaluate the integral : `\int \frac{sec^2x}{\sqrt{25tan^2x}}dx` Let `tanx =t` So, `sec^2x dx=dt` Therefore we have, `\int \frac{sec^2x}{\sqrt{25tan^2x}}dx=\int...

MathGiven the integral: `\int \frac{2}{x\sqrt{9x^225}}dx` Let `x=\frac{5}{3}sect` `` So, `dx=\frac{5}{3}sect tant dt` Hence we have, `\int \frac{2}{x\sqrt{9x^225}}dx=\int \frac{\frac{10}{3}sect...

MathThe substitution `u = 1/2 e^(2x)` make this integral a wellknown one: `du = e^(2x) dx,` so `e^(2x) dx` from the numerator is equal to `du,` and `4 + e^(4x)` from the denominator is equal to `4 +...

MathWe have to evaluate the integral: `\int \frac{1}{x\sqrt{1(ln x)^2}}dx` Let `ln x=u` So, `\frac{dx}{x}=du` Therefore we have, `\int \frac{1}{x\sqrt{1(ln(x))^2}}dx=\int \frac{du}{\sqrt{1u^2}}`...

MathMake the substitution `u = t^2/5,` then `du = (2t dt)/5,` `t dt = 5/2 du` and `t^4 = 25u^2.` The integral becomes `int (5/2)/(25u^2 + 25) du = 1/10 int (du)/(1 + u^2) = 1/10 arctan(u) + C =...

MathFor the given integral: `int 1/(xsqrt(x^44))dx` , we may apply usubstitution by letting: `u =x^44 ` then ` du = 4x^3 dx` . Rearrange `du = 4x^3 dx` into `(du)/( 4x^3)= dx` Plugin `u =x^44`...

MathIndefinite integral are written in the form of int `f(x) dx = F(x) +C` where: f(x) as the integrand F(x) as the antiderivative function C as the arbitrary constant known...

MathGiven the integral : `\int \frac{1}{4+(x3)^2}dx` let `x3=t` So, `dx=dt` therefore we have, `\int \frac{1}{4+(x3)^2}dx=\int \frac{1}{4+t^2}dt` `=\int \frac{1}{2^2+t^2}dt`...

MathIndefinite integral are written in the form of `int f(x) dx = F(x) +C` where: f(x) as the integrand F(x) as the antiderivative function C as the arbitrary constant known...

MathRecall that the indefinite integral is denoted as: `int f(x) dx =F(x)+C` There properties and basic formulas of integration we can apply to simply certain function. For the problem `int...

MathWe have to evaluate the integral : ```\int \frac{dx}{x\sqrt{4x^21}}` Let `\sqrt{4x^21}=u` So, `\frac{1}{2\sqrt{4x^21}}.8x dx=du` `\frac{4xdx}{\sqrt{4x^21}}=du`...

MathWe have to evaluate `\int \frac{dx}{\sqrt{14x^2}}` Let `x=\frac{1}{2} sint ` So, `dx= \frac{1}{2}cost dt` Hence we have, `\int \frac{dx}{\sqrt{14x^2}}=\int \frac{\frac{1}{2}cost...

MathWe have to evaluate the integral: `\int \frac{dx}{\sqrt{9x^2}}` let `x=3sint` So, `dx=3cost dt` Hence we have, `\int \frac{dx}{\sqrt{9x^2}}=\int \frac{3cost}{\sqrt{99sin^2t}}dt`...

MathFirst, check that the given point satisfies the equation: `arctan(1 + 0) = 0 + pi/4` is true. The slope of the tangent line is `y'(x)` at the given point. Differentiate the equation with respect to...
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