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• Math
First, draw the triangle formed by the three equations x+y=1, x=1 and y=1. Let the vertices of the triangle be A, B and C (see attached figure). Base on the graph, the coordinates of the vertices...

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• Math
In order to answer this question, let's take a look at the reasoning behind it! There are many different kinds of probability problems, but this one relates to the Factorial Rule, which is: n! = n...

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• Math
Hello! This figure is really a parallelogram, for example because the opposite sides have the same length: `|AB| = |CD| = 2sqrt(2)` and `|BC| = |AD| = 2sqrt(37).` Or we can check that the opposite...

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• Math
Hello! The most clear method to solve this is to denote the variables and to solve the system of equations that occurs. This system is simple! Let `S` be the amount of dollars Soumik have, and `K`...

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• Math
Hello! Denote the slope of this line as `m.` The vertical line (which has an undefined slope) doesn't suit us, so we'll not miss a solution. Horizontal line with `m=0` doesn't suit also, so we can...

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• Math
We 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...

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• Math
In 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...

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

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• Math
Hello! 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`...

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• Math
Denote 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 +...

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• Math
Here 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...

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• Math
Hello! 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...

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• Math
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...

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• Math
The 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.)...

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

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• 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 theta-int d theta=` Since the second integral is easy `int d theta=theta+C` we...

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• Math
`int(2-tan(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 theta-int tan(theta/4)d theta=` Since the first integral is easy `int 2d...

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• 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...

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

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• Math
`int (sec(x)tan(x))/(sec(x)-1)dx=` 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 the numerator is...

Asked by enotes on via web

• 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,`...

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• Math
Let's make the substitution `y = sqrt(x),` then `dy = (dx)/(2sqrt(x))` and the integral becomes `int (dx)/(sqrt(x)(1-3sqrt(x))) = int (2 dy)/(1-3y),` Using log integrations rules we find that it is...

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• 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^(1/3-1)dx` `du=1/3x^(-2/3)dx` `du=dx/(3x^(2/3))`...

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• Math
Let's make a substitution `y = x-1,` 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 ln|y| - 2/y + C = 2 ln|x-1| - 2/(x-1) + C,` where...

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• Math
Find the indefinite integral -- `int (x(x-2))/(x-1)^3 dx ` : Rewrite the integral using partial fractions and integrate term by term: `=int ( 1/(x-1)-1/(x-1)^3)dx ` `=int 1/(x-1)dx - int...

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• Math
Solving for indefinite integral using u-substitution 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...

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• Math
Well, let's `u = 1 + sqrt(3x),` then `x = (u-1)^2/3` and `dx = 2/3 (u-1) du.` So the integral becomes `int 1/u * 2/3 * (u-1) du = 2/3 int(1 - 1/u) du = 2/3 (u - ln|u|) + C = 2/3 (1+sqrt(3x) -...

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• Math
To apply u-substitution , 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...

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• Math
Solving indefinite integral by u-substitution, 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,...

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• Math
`intcot(theta/3)d(theta)` `intcotudu=ln|sinu|+C` Let `u=theta/3` `(du)/[d(theta)]=1/3` `d(theta)=3du` `intcot(theta/3)d(theta)` `=intcot(u)[3du]` `=3intcot(u)du` `=3ln|sin(u)|+C`...

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

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• Math
`intcsc(2x)dx` `intcsc(u)du=ln|csc(u)-cot(u)|+C` Let `u=2x` `(du)/(dx)=2` `dx=1/2du` `intcsc(2x)dx` `=intcsc(u)(1/2du)` `=1/2intcsc(u)(du)` `=1/2ln|csc(u)-cot(u)|+C`...

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• Math
`int (x^2+2x+3)/(x^3+3x^2+9x)dx=` 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 the numerator is...

Asked by enotes on via web

• Math
`int(x^2+4x)/(x^3+6x^2+5)dx=` 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 the numerator is equal...

Asked by enotes on via web

• Math
` int (x^2-3x+2)/(x+1)dx ` : Find the indefinite integral: Rewrite the integrand using long division or synthetic division and integrate term by term: `int (x^2-3x+2)/(x+1)dx=int (x-4+6/(x+1))dx `...

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• Math
`int (2x^2+7x-3)/(x-2)dx` To solve, divide the numerator by the denominator (see attached figure). `= int (2x + 11 + 19/(x-2)) dx` Express it as sum of three integrals. `= int 2xdx + int11dx + int...

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• Math
`int (x^3-3x^2+5)/(x-3)dx` To solve, divide the numerator by the denominator. `= int (x^2 + 5/(x-3))dx` Express it as sum of two integrals. `= int x^2dx + int 5/(x-3)dx` For the first integral,...

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• Math
`int(x^3-6x-20)/(x+5)dx` Let's evaluate the integral by applying integral substitution, Let u=x+5, `=>x=u-5` du=dx `int(x^3-6x-20)/(x+5)dx=int((u-5)^3-6(u-5)-20)/udu`...

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• Math
In 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...

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• Math
`int (x^3-4x^2-4x+20)/(x^2-5)dx` To solve, divide the numerator by the denominator (see attached figure). `= int (x - 4 + x/(x^2-5))dx` `= int xdx - int4dx + int x/(x^2-5)dx` For the first...

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

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• Math
First, 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 ln|y| + C = 1/3 ln|lnx| + C.` This...

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• Math
`int (x^2-4)/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...

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• Math
`int (x^3-8x)/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...

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• Math
Let's make a substitution y = 2x+5. Then x = (y-5)/2, dx = 1/2 dy. The integral becomes `int 1/y * 1/2 dy = 1/2 ln|y| + C = 1/2 ln|2x+5| + C.` This is the answer.

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• Math
`int 9/(5-4x)dx=` Use homogeneity of integral: `int alpha f(x)dx=alpha int f(x)dx,` `alpha in RR` `9int dx/(5-4x)=` Use substitution: `u=5-4x,` `du=-4dx=>dx=-(du)/4.` `9int(-du)/(4u)=` Use...

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• Math
`int x/(x^2-3)dx=` Make substitution: `u=x^2-3,` `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...

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

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

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