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MathBefore evaluating the definite integral, to determine the area located under the curve, the given lines and x axis, we verify if the function is an even function. A function is even if f(x) =...

MathTo rewrite 1/(x^3+x^2+x+1) as partial fractions, we first need to factorize the denominator 1/(x^3+x^2+x+1) => 1 / x^2(x + 1) + 1(x +1) => 1 / (x^2 + 1)(x + 1) Let this be equal to (Ax +B) /...

MathThe answer is very simple one. Because the given function is continuous and odd. But let's see how it works. We know that if a function is continuous over a range [a ; a], then the following...

MathYou have to solve the equation 4(2y+1) = 2(12y). First let's start with 4(2y+1) = 2(12y) open the brackets => 4*2y + 4*1 = 2*12  2*y => 8y + 4 = 24  2y now move the terms with y to one...

MathGiven the data from enunciation, we conclude that the domain of definition of the function is the closed interval [a ; a]. f(x):[a;a]>R If f is a continuous function over the closed interval,...

MathWe have a = (2^(2n+3)*3^(n+2)*5^(2n+1) + 3^(n+3)*4^(n+2)*5^(2n))/792 => (2^3*2^2n*3^2*3^n*5^1*5^2n + 3^n*3^3*4^n*4^2*5^2n)/792 => (8*9*5*2^2n*3^n*5^2n + 27*16*3^n*4^n*5^2n)/792 =>...

MathWe have to solve the equation 0.2*[0.2*(x0.2)0.2]0.2 = 0.2 0.2*[0.2*(x  0.2)  0.2]  0.2 = 0.2 divide all the terms by 0.2 => 0.2*[0.2*(x  0.2)  0.2]/ 0.2  0.2/0.2 = 0.2/02. =>...

MathIt is given that the family has eaten 1.5 kg on the first day and the next day they ate 0.35 kg more than they did on the first day. So they ate 1.5 + 0.35 = 1.85 kg on the second day. If we add...

MathSo, we'll have to prove that: Int x*f(x)dx = (a+b)/2*Int f(x)dx We'll calculate the integral from the left side substituting x = a + b  t. By definition, f(a + b  t) = f(t), so the integral will...

MathThe question states that f(x) = 1/ (cos x)^4. We have to find the integral of f(x) = 1/ (cos x)^4. (cos x)^4 = (sec x)^4 = (sec x)^2 * (sec x)^2 => ((tan x)^2 + 1)* (sec x)^2 => ((tan...

MathThe question gives f(x)=x^3+ln(x) and g(x)=1/x+3x^2. Differentiating f(x), we use the relations that the derivative of x^n is n*x^(n1) and the derivative of ln x is 1/x. These five f'(x) = 3*x^2 +...

MathThe question states that (sin y)^2 = 5* sin y  6. let x = sin y => x^2 = 5x  6 => x^2  5x + 6 = 0 => x^2  3x  2x + 6 = 0 => x(x 3)  2( x  3) =0 => (x  2) (x 3) = 0 x has...

MathGiven the function: y= 3x^2 + 2x + 5 To find the extreme values, first we will find the derivative y'. ==> y' = 6x + 2 Now we will find the critical values which is the derivatives zeros....

MathFor the given lines , 2xy+2=0 and x+y4=0, we first find their slope. 2x  y + 2 = 0 => y = 2x  2 => y = 2x + 2 This is in the form y = mx + c , where m is the slope of the line. So the...

MathLet f(x) = sin^4 (x^2 +1) We need to find f'(x). We will use the chain rule to find the derivative. Let f(x) = u^4 such that: u= sin(x^2 +1) ==> f'(x) = 4u^3 * (u') Now we will calculate u'....

MathThe problem states that (x+x+....+x)*1.8 = 3.6, the bracket has x written 10 times. Adding x ten times is the same as multiplying it by 10, so we can write the equation as: 10*x*1.8 = 3.6 => 18x...

MathThe arithmetic mean of a and b is given as (a + b)/2 Here x= (6.64+2.36):(4.51.5) => x = 9 / 3 => x = 3 y=(196.4):(4.9+1.4) => y = 12.6 / 6.3 => y = 2 The arithmetic mean of 3 and 2...

MathWe have to determine the set of natural numbers in which n lies given that : 2/5=<(n+1)/10=<1/2 2/5=<(n+1)/10=<1/2 multiply all the terms by 10 => 4 =< (n +1) =< 5 subtract 1...

MathTo find the indefinite integral of f(x)=x^2(x^3+1)^4, let us first denote x^3 + 1 = t => dt/dx = 3x^2 => x^2 dx = (1/3) dt Int [ x^2(x^3+1)^4 dx] => Int [ (1/3) t^4 dt] => t^5 / (3*5)...

MathWe have to find a given that 12*a + 21*a + 23*a + 32*a + 22*a = 55. First add all the terms with a => (12 + 21 + 23 + 32 + 22)*a = 55 => 110*a = 55 divide both sides by 110 => a = 55/110...

MathWe have to prove that the fraction (3^3+3^4+3^5)/(5^5+5^7) is divisible by 13. (3^3+3^4+3^5)/(5^5+5^7) factor 3^3 in the numerator => 3^3( 1 + 3 + 3^3) / (5^5+5^7) => 3^3 ( 1 + 3 + 9) /...

Math(6n+15)/3n = K where k is a natural number ( 1,2,3,4...) Let us simplify. ==> 6n/3n + 15/3n = k ==> 2 + 5/n = k We have 2 is a natural number. Then 5/n must we a natural number too. ==>...

Math(3+x)/5 < 1 We need to find x such that x is a natural number. ==> We will solve the inequality. ==> We will multiply by 5. ==> 3+ x < 5 Now we will subtract 3 from both sides....

MathWe have 2^x = 16 To solve, first e will rewrite the numbers so the bases are equal. We will factor 16. ==> 16 = 4*4 = 2*2*2*2 = 2^4 Now we will substitute into the equation. ==> 2^x = 2^4 Now...

MathFor example, we have the points A(2,5) and the point B(4,3) and we need to find the point in between. We will average both x and y coordinates to find the midpoint. Let the midpoint be M(xM, yM)....

MathGiven the functions: f(x) = 3x5 g(x) = x^2 We need to find fog(x). First, we will determine the function fog(x). ==> fog(x) = f(g(x)) We will substitute with g(x) = x^2 => fog(x) = f (x^2)...

MathGiven the function: f(x) = ax^2  5x +10 We need to find the value of a so the function has one real root. We know that if f(x) has one root, then delta must be zero. ==> delta = b^2 4ac = 0...

MathGiven the system: 2x y = 5...........(1) x+3y = 12..............(2) Then, we have a system of two equations and two variables. We will use the elimination method to solve for x and y. We will...

MathGiven f(x) = (x3) / ln x We need to find the derivative of f(x). We notice that f(x) is a quotient of two functions. Then, we will use the quotient rule to find f'(x). ==> Let f(x) = u/v such...

MathGiven the inequality: 2x 3 < x+5 We need to find all x values that verifies the inequality. First we will combine like terms. ==> We will subtract x from both sides. ==> 2x x 3 < x+5...

MathIf math would be a color, shape, and sound. What would it be?If math would be a color, shape, and...This is a very intriguing question that offers the student the opportunity to really think "outside the box." But, it is also a subjective question that prompts the creative and emotional side of...

MathTh e cone has the vertical height 30 and radius 10. Si the semi vertical angle x of the cone is given by tanx radiuus/ height . Or tan x = 10/30 = 1/3. Let the cylinder describes in side the cone...

MathWe have to find the integral of y=(4+ln x)^3/x. To start, let us take t = ln x => dt/dx = 1/x => dx/x = dt Int [y] = Int [(4+ln x)^3/x dx] => Int [ (dx/x) ( 4 + ln x)] => Int [ (4 +...

MathThe polynomial with the roots 2, 1 sqrt 3 and 1 + sqrt 3 is: (x +2)( x  1 + sqrt 3)(x  1  sqrt 3) => (x + 2)((x  1)^2  3) => (x + 2)( x^2 + 1  2x  3) => (x + 2)(x^2  2x  2)...

MathTo determine the point of maxima for the curve y = x^2  8x + 16, we need to find the derivative of y and equate it to zero. y = x^2  8x + 16 => y' = 2x  8 2x  8 = 0 => x = 8/2 => x =...

MathLet the two numbers we have to find the squares of be A and B. Now their sum is A + B = 3 Their product is A*B = 2 We need A^2 + B^2. This can be written as (A + B)^2  2A*B => 3^2  2*2 => 9...

MathTo find the area under the curve y = cos x*sin 5x and between the lines x = 0 and x= pi, is the definite integral of y between x = 0 and x = pi. Now Int [y] = Int [ cos x*sin 5x dx] =>  (cos...

MathWe have to differentiate f(x) = [cos (x^3) ]^3 We have to use the chain rule for three functions x^3, cos x and x^3. This gives: f'(x) = 3*[cos (x^3) ]^2 *(sin x^3)* 3x^2. => 9* [cos (x^3) ]^2...

MathWe have to find the anti derivative of f(x) = 1/(1+x^2)*arc tan x Here we first take t = arc tan x => dt/dx = 1 / ( 1 + x^2) => dt = dx / ( 1 + x^2) Int [ 1/(1+x^2)*arc tan x dx] => Int [...

MathWrite a sine and a cosine equation to model the movement of the boat in the following case:A boat...The vertical distance between the high and the low points of the boat is 9 m. This means that the high point is 4.5 m above the mean and the low point is 4.5 m below the mean. And there are 5...

MathWe have the two equations: x + 2y = 5 … (1) 2x + 4y = 6 … (2) From (1) we have x + 2y = 5 => 2x + 4y = 10 And from (2) we have 2x + 4y = 6. Both of these cannot be true as it would imply 6 =...

MathYou have to use imaginary numbers here. Remember that i is the square root of 1. Since  9 is 1*9, then the square root of 9 is the square root of 1 times the square root of 9. This means that...

MathThis might be better suited for the Math board, but I will give it a shot. Since a cube has six sides we can first divide the 194,4 by 6. This will us give us the area of each side, then we know...

MathTo find if the two curves intercept or not, we equate them, as at the point of intersection the x and y coordinates are the same. x^23x+1 = 2x^2+x+4 => 2x^2  x^2 + x + 3x + 4  1 = 0 => x^2...

MathProve Sin A + Sin B + Sin C  Sin(A+B+C) = 4 Sin ((A+B)/2) Sin ((B+C)/2) Sin ((C+A)/2). Given A+B+C = 180 degree. We use sin x+sin y = 2sin(x+y)/2 *cos(xy)/2. So Sin A+sin B = 2 sin (A+B)/2 *...

MathFirst we find the integral of y=cos x/(4+sin x) Let 4 + sin x = t => dt/dx = cos x => dt = cos x dx Int [ cos x/(4+sin x) dx] => Int [ (1/t) dt] => ln t + C => ln ( 4 + sin x) + C...

MathWe have to integrate y = 1 / (x2)^1/3 Int [ y ] = Int [ 1 / (x2)^1/3 dx] Let x  2 = t dt/dx = 1 => dt = dx Int [ 1 / (x2)^1/3 dx] => Int [ 1 / t^1/3 dt] => Int [ t^(1/3) dt] =>...

MathWe have to solve 4^(x+1)/2 = 12  2x. 4^(x+1)/2 = 12  2x => 4^(x+1) = 24  4x => 4^x*4^1 = 24  4x => 4^x = 24/4  4x/4 => 4^x = 6  x It is not possible to solve any further for x.

MathWe have to write (3x2)/(x3)(x+1) as the sum of fractions. Let us write (3x2)/(x3)(x+1) = A / (x  3) + B/(x +1) => [A(x +1) + B(x  3)]/(x3)(x+1) = (3x2)/(x3)(x+1) => Ax + A + Bx  3B...

MathWe have to find the solution x for the equation cos x + cos 3x = 0. Using the relation cos 3x = 4 ( cos x)^3  cos x, we can write cos x + cos 3x = 0 as: cos x + 4 ( cos x)^3  cos x = 0 => 4 (...