
MathWe are given a+b+c=395, b+c+d=1001 and a<b<c<d and we are asked to find the value of d: As stated, d can take on a range of values. Subtract the two equations: b+c+d=1001a+b+c...

MathIn order to find the greatest possible value of the highest number, we must consider the lowest possible values of five of the numbers. We are given the information that all the numbers are...

MathGiven ` x^15x^13+x^11x^9+x^7x^5+x^3x=7 ` , we are asked to show that `x^16>15 ` : First, note that `x^15x^13+x^11x^9+x^7x^5+x^3x=x(x1)(x+1)(x^4+1)(x^8+1) ` so the polynomial has real...

MathHello! As I understand, the deceleration is uniform (the same all the time). Denote it as `agt0` and denote the initial speed as `V_0.` In m/s `V_0 = 140/3.6.` Then the speed is `V(t) = V_0  a*t`...

MathDenote the numbers as `a_1 lt= a_2 lt= a_3 lt= a_4 lt= a_5 lt= a_6 lt= a_7.` It is given that: `(a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7)/7 = 12,` `(a_1 + a_2 + a_3 + a_4)/4 = 8,` `(a_4 + a_5 +...

MathHere are some areas of interest and questions that arise in mathematics teaching in the elementary grades: Curriculum  What content should be taught? When should it be taught? What are the...

MathHello! This expression is already a sum of two numbers, `sin(32)` and `sin(54).` Probably you want or express it as a product, or as an expression involving trigonometric functions of sum or...

MathThe perimeter is 46 meters and its length is 2 meters more than twice its width. What is the length?In this problem, the length is compared to the width of the rectangle. So let's assign a variable that represents the width of the rectangle. Let the width be w. `width = w` Since the length is 2...

MathThe definition of probability is the number of occurrences that meet a specified criteria (the size of the event space) divided by the total number of possibilities (the size of the sample space.)...

Math`int(sec(2x)+tan(2x))dx=` Use additivity of integral: `int (f(x)+g(x))dx=int f(x)dx+int g(x)dx.` `int sec(2x)dx+int tan(2x)dx=` Make the same substitution for both integrals: `u=2x,`...

Math`int (sec(x)tan(x))/(sec(x)1)dx=` We will use the following formula: `int (f'(x))/(f(x))dx=lnf(x)+C` The formula tells us that if we have integral of rational function where the numerator is...

Math`int (csc^2(t))/(cot(t))dt=` We will use the following formula: `int (f'(x))/(f(x))dx=lnf(x)+C.` We will use the following formula: The formula tells us that if we have integral of rational...

Math`int (cos(t))/(1+sin(t))dt=` We will use the following formula: `int (f'(x))/(f(x))dx=ln(f(x))+C.` The formula tells us that if we have integral of rational function where numerator is equal to...

Math`int(2tan(theta/4))d theta=` Use additivity of integral: `int(f(x)pm g(x))dx=int f(x)dx pm int g(x)dx.` `int2d thetaint tan(theta/4)d theta=` Since the first integral is easy `int 2d...

Math`int (cos(3theta)1)d theta=` Use additivity of integral: `int (f(x)+g(x))dx=int f(x)dx+int g(x)dx.` `int cos(3theta)d thetaint d theta=` Since the second integral is easy `int d theta=theta+C` we...

Math`intsec(x/2)dx` `sec(u)du=lnsec(u)+tan(u)+C` Let `u=x/2` `(du)/(dx)=1/2` `dx=2du` `intsec(x/2)dx` `=intsec(u)(2du)` `=2intsec(u)(du)` `=2lnsec(u)+tan(u)+C`...

Math`intcsc(2x)dx` `intcsc(u)du=lncsc(u)cot(u)+C` Let `u=2x` `(du)/(dx)=2` `dx=1/2du` `intcsc(2x)dx` `=intcsc(u)(1/2du)` `=1/2intcsc(u)(du)` `=1/2lncsc(u)cot(u)+C`...

Math`inttan(5theta)d(theta)` `tan(u)du=lnsec(u)+C ` Let `u=5theta` `(du)/[d(theta)]=5` `d(theta)=(du)/5` `inttan(5theta)d(theta)` `=inttan(u)[(du)/5]` `=1/5inttanu(du)` `=1/5lnsec(u)+C`...

Math`intcot(theta/3)d(theta)` `intcotudu=lnsinu+C` Let `u=theta/3` `(du)/[d(theta)]=1/3` `d(theta)=3du` `intcot(theta/3)d(theta)` `=intcot(u)[3du]` `=3intcot(u)du` `=3lnsin(u)+C`...

MathSolving indefinite integral by usubstitution, we follow: `int f(g(x))*g'(x) = int f(u) *du` where we let `u = g(x)` . By following the instruction to let "u" be the denominator of the integral,...

MathTo apply usubstitution , we let `u = sqrt(x)3` . Then ` du = 1/(2sqrt(x) dx` . Rearrange `du = 1/(2sqrt(x)) dx` into `dx =2sqrt(x) du` Substituting `dx=2sqrt(x) du` and `u =sqrt(x)3` : `int...

MathWell, let's `u = 1 + sqrt(3x),` then `x = (u1)^2/3` and `dx = 2/3 (u1) du.` So the integral becomes `int 1/u * 2/3 * (u1) du = 2/3 int(1  1/u) du = 2/3 (u  lnu) + C = 2/3 (1+sqrt(3x) ...

MathSolving for indefinite integral using usubstitution follows: `int f(g(x))*g'(x) dx = int f(u) du` where we let` u = g(x)` . In this case, it is stated that to let u be the denominator of integral...

MathFind the indefinite integral  `int (x(x2))/(x1)^3 dx ` : Rewrite the integral using partial fractions and integrate term by term: `=int ( 1/(x1)1/(x1)^3)dx ` `=int 1/(x1)dx  int...

MathLet's make a substitution `y = x1,` then `dy = dx` and `x = y+1.` The integral becomes `int (2(y+1))/y^2 dy = 2 int (dy)/y + 2 int (dy)/y^2 = 2 lny  2/y + C = 2 lnx1  2/(x1) + C,` where...

Math`int1/(x^(2/3)(1+x^(1/3)))dx` Evaluate the integral by applying integral substitution, Let `u=1+x^(1/3)` `du=1/3x^(1/31)dx` `du=1/3x^(2/3)dx` `du=dx/(3x^(2/3))`...

MathLet's make the substitution `y = sqrt(x),` then `dy = (dx)/(2sqrt(x))` and the integral becomes `int (dx)/(sqrt(x)(13sqrt(x))) = int (2 dy)/(13y),` Using log integrations rules we find that it is...

MathFirst, note that `ln(x^3) = 3ln(x).` Then make a substitution `y = ln(x),` thus `dy = dx/x.` So the integral becomes `int (dx)/(3x ln(x)) = 1/3 int (dy)/y = 1/3 lny + C = 1/3 lnlnx + C.` This...

Math`int (ln(x))^2/xdx` To solve, apply usubstitution method. Let, `u= ln x` Then, differentiate it. `du=1/xdx` Plugin them to the integral. `int (ln(x))^2/xdx` `= int (ln(x))^2 * 1/xdx` `=int u^2...

Math`int (x^34x^24x+20)/(x^25)dx` To solve, divide the numerator by the denominator (see attached figure). `= int (x  4 + x/(x^25))dx` `= int xdx  int4dx + int x/(x^25)dx` For the first...

MathIn Substitution Rule, we follow` int f(g(x))g'(x) dx = int f(u) du ` where we let `u = g(x)` . Before we use this, we look for possible way to simplify the function using math operation or...

Math`int(x^36x20)/(x+5)dx` Let's evaluate the integral by applying integral substitution, Let u=x+5, `=>x=u5` du=dx `int(x^36x20)/(x+5)dx=int((u5)^36(u5)20)/udu`...

Math`int (x^33x^2+5)/(x3)dx` To solve, divide the numerator by the denominator. `= int (x^2 + 5/(x3))dx` Express it as sum of two integrals. `= int x^2dx + int 5/(x3)dx` For the first integral,...

Math`int (2x^2+7x3)/(x2)dx` To solve, divide the numerator by the denominator (see attached figure). `= int (2x + 11 + 19/(x2)) dx` Express it as sum of three integrals. `= int 2xdx + int11dx + int...

Math` int (x^23x+2)/(x+1)dx ` : Find the indefinite integral: Rewrite the integrand using long division or synthetic division and integrate term by term: `int (x^23x+2)/(x+1)dx=int (x4+6/(x+1))dx `...

Math`int(x^2+4x)/(x^3+6x^2+5)dx=` We will use the following formula: `int(f'(x))/(f(x))dx=lnf(x)+C` The formula tells us that if we have integral of rational function where the numerator is equal...

Math`int (x^2+2x+3)/(x^3+3x^2+9x)dx=` We will use the following formula: `int (f'(x))/(f(x))dx=lnf(x)+C` The formula tells us that if we have integral of rational function where the numerator is...

Math`int (x^38x)/x^2dx` To solve, express the integrand as two fractions with same denominators. `=int (x^3/x^2(8x)/x^2)dx` Simplify each fraction. `=int (x  8/x)dx` Express it as difference of two...

Math`int (x^24)/xdx` To solve, express the integrand as two fractions with same denominators. `=int (x^2/x  4/x)dx` Simplify the fractions. `= int (x 4/x) dx` Express it as difference of two...

Math`int (4x^3+3)/(x^4+3x)dx` To solve, apply usubstitution method. So let: `u= x^4+3x` Then, differentiate it. `du=(4x^3+3)dx` Plugin them to the integral. `int (4x^3+3)/(x^4+3x)dx` `= int...

Math`intx^2/(5x^3)(dx)` Let `u=5x^3` `(du)/(dx)=3x^2` `(dx)=(du)/(3x^2)` `intx^2/u*(du)/(3x^2)` `=(1/3)int(1/u)du` `=(1/3)lnu+C` `=(1/3)ln5x^3+C` The final answer is:...

Math`int x/(x^23)dx=` Make substitution: `u=x^23,` `du=2xdx=>xdx=(du)/2.` `int (du)/(2u)=` Use homogeneity of integral: `int alpha f(x)dx=alpha int f(x)dx,` `alpha in RR.` `1/2int(du)/u=1/2ln...

Math`int 9/(54x)dx=` Use homogeneity of integral: `int alpha f(x)dx=alpha int f(x)dx,` `alpha in RR` `9int dx/(54x)=` Use substitution: `u=54x,` `du=4dx=>dx=(du)/4.` `9int(du)/(4u)=` Use...

MathLet's make a substitution y = 2x+5. Then x = (y5)/2, dx = 1/2 dy. The integral becomes `int 1/y * 1/2 dy = 1/2 lny + C = 1/2 ln2x+5 + C.` This is the answer.

Math`int1/(x5)(dx)`` ` `u=x5 ` `(du)/(dx)=1` `(du)=(dx)` `int1/u(du)` `=lnu+C` `lnx5+C` The final answer is: `lnx5+C` ``

Math`int1/(x+1)(dx)` Let `u=x+1` `(du)/(dx)=1` `(du)=(dx)` `int1/u(du)` `=lnu+C` `=lnx+1+C` The final answer is: lnx+1+C

Math`int10/x(dx)` `=10int1/x(dx)` `=10lnx+C` The answer is: `10lnx+C`

Math`int5/x(dx)` `=5int1/x(dx)` `=5lnx+C` The answer is: `5lnx+C`

MathHello! Let's differentiate the given equation: `((x+y(x))^2)' = 4',` `2(x+y(x))(x+y(x))' = 0,` `2(x+y)(1+y') = 0.` We know that `(x+y)^2 = 4,` therefore `x+y = +2` and it is never zero. Thus...

MathHello! By the definition, the average speed is the total distance traveled divided by the total time spent. In our problem the total distance and the total time consist of the two halves. Denote...
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