
Math
You need to evaluate the limit, hence, you need to replace 0 for x in expression under the limit, such that: `lim_(x>0) (e^(x)  1)/(tan x) = (e^0  1)/(tan 0) = (11)/0 = 0/0` Hence, since the...

Math
You need to evaluate the limit, hence, you need to replace 0 for x in expression under the limit, such that: `lim_(x>0) (tan 4x)/(x + sin 2x) = (tan 0)/(0 + sin 0) = 0/0` Hence, since the...

Math
You need to evaluate the limit, hence, you need to replace 0 for x in expression under the limit, such that: `lim_(x>0) (e^(4x)  1  4x)/(x^2) = (e^0  1  0)/(0) = (11)/0 = 0/0` Hence, since...

Math
You need to evaluate the limit, hence, you need to replace `oo` for x in expression under the limit, such that: `lim_(x>oo) (e^(4x)  1  4x)/(x^2) = (e^oo 1  oo)/(oo) = (oo)/(oo)` Hence,...

Math
We are given that f(x)=sqrt(x) and g(x) = x+1: We are asked to find the composition of the functions: (1) fog(x)=f(g(x)) We substitute the expression for g(x) for x in f(x). f(g(x))=sqrt(x+1) (2)...

Math
The slope of the tangent line to the curve parallel to the xaxis will be the point(s) where the first derivative is zero: f(x)=x^3+3x+14 f'(x)=3x^2+3 f'(x)=0 ==> 3x^2+3=0 3x^2=3 x^2=1 There...

Math
Let the numerator be `x` , then the denominator becomes `x+1.` `therefore ` the fraction = `x/(x+1)` According to the question, 1 is added to the numerator and 5 is added to denominator,i.e. `x+1...

Math
Hello! The properties, or laws, that can be applied here, are (`a,` `b,` `c` denote any integer, real or complex numbers): 1a) The Commutative Law of Addition: `a+b=b+a` 1b) The Commutative Law of...

Math
The difference of 3 times a number 5. This is stated however I wonder if there is supposed to be an "and" 5. So first thing you want to do is define a variable. Let x = the number, therefore the...

Math
To multiply fractions, we have to multiply the top numbers and multiply the bottom numbers. And, cancel the common factor to simplify it. For example: `4/9 xx 6/7` Multiply the numerators and...

Math
Hello! The exponential function `f` has a form `f(x)=a*b^x` `(bgt0).` To find `a` and `b,` substitute given x'es. The first point, x=0 and f(x)=2100: `2100 = f(0) = a*b^0 = a.` So `a = 2100.`...

Math
Hello! For real number `x,` the symbol `[x]` means the integer part of `x,` i.e. the largest integer number not greater than `x.` In decimal notation for positive `x=n.mkl...,` `[x]=n.` For...

Math
I assume we are talking about the chromatic number of a graph G  each vertex of the graph is "colored" (numbered, assigned a letter, etc...) such that any connected vertices have a different...

Math
The cost for 100g of bird seed is 40p. Sasha has paid 2 pounds or 200p for her birdseed. 100g/40p=?g/200p Let x represent the number of grams bought: (200)(100)/40=x x=500...

Math
We need a set of numbers whose mean is 65 and whose m.a.d. (mean absolute deviation) is 6.5. Suppose the set has 8 numbers. Then the sum of the absolute values of the differences of the values and...

Math
x = prime number y = prime number x+y = 85 1, 84 (neither are prime) 2, 83 (both are prime) 3, 82; 4, 81; 5, 80; 6, 79, ... all evens are paired with a possible odd prime therefore both won't be...

Math
Assuming color represents the relations input and the number of students represents the output, the relations inverse would not be a function. To determine switch the inputs and outputs of a...

Math
To prove that `n^4  m^4n^2  m^2 + 2` is always even, group the terms containing the same variables together: `n^4  m^4  n^2  m^2 + 2 = n^4  n^2  m^4  m^2 + 2` Consider each group...

Math
Hello! Let's denote `a_n` = the quantity of ndigit numbers in which every digit is a 6 or a 7 and no two 7s are adjacent. We are interested in `a_7` . Also, denote `b_n` = the quantity of ndigit...

Math
Find `sum_(n=4)^6 6n` This is the sum 6(4)+6(5)+6(6)=90 which is the answer. An alternative is to "factor" out the 6 and you get 6(4+5+6)=6(15)=90.

Math
`f''(x)=x^(3/2)` `f'(x)=intx^(3/2)dx` `f'(x)=x^(3/2+1)/(3/2+1)+C_1` `f'(x)=2x^(1/2)+C_1` `f'(x)=2/sqrt(x)+C_1` Now , solve for C_1 , given f'(4)=2 `2=2/sqrt(4)+C_1` `2=1+C_1` `C_1=3`...

Math
`f''(x)=sin(x)` `f'(x)=intsin(x)dx` `f'(x)=cos(x)+C_1` Let's find constant C_1 , given f'(0)=1 `f'(0)=1=cos(0)+C_1` `1=1+C_1` `C_1=2` `:.f'(x)=cos(x)+2` `f(x)=int(cos(x)+2)dx`...

Math
`int(1csc(c)cot(t))dt` apply the sum rule, `=int1dtintcsc(t)cot(t)dt` `=tint((1/sin(t)(cos(t)/sin(t)))dt` `=tintcos(t)/(sin^2(t))dt` Now, let sin(t)=x `rArr` cos(t)dt=dx `=tintdx/x^2` apply...

Math
It is `int(t^2)dt+int(sec^2(t))dt=t^3/3 + tan(t)+C.` where C is any constant. Both integrals are from must know table.

Math
You need to evaluate the indefinite integral, such that: `int(sec^2 theta  sin theta) d theta = int sec^2 theta d theta  int sin theta d theta` `int(sec^2 theta  sin theta) d theta = int...

Math
You need to evaluate the indefinite integral, such that: `int sec y(tan y  sec y) dy= int (sin y  1)/(cos^2 y) dy = int (sin y)/(cos^2 y) dy  int 1/(cos^2 y) dy` You need to solve `int (sin...

Math
You need to evaluate the indefinite integral and yo need to use the following trigonometric formula `1 + tan^2 y = 1/(cos^2 y).` Replacing `1/(cos^2 y)` for `1 + tan^2 y` yields: `int (1 + tan^2 y)...

Math
You need to evaluate the indefinite integral, such that: `int(4x  csc^2 x) dx= int 4x dx int csc^2 x dx` `int(4x  csc^2 x) dx= int 4x dx int 1/(sin^2 x) dx` You need to remember that `1/(sin^2...

Math
To find the solution of the differential equation `f'(x)=6x` ` ` with initial condition `f(0)=8 ` , integrate both sides: `int(f'(x) dx)=int(6x dx) =>f(x)+c_1=3x^2+c_2` ,where `c_1,c_2 ` are...

Math
You need to use direct integration to evaluate the general solution to the differential equation: `int (4x^2)dx = 4x^3/3 + c` You need to find the particular solution using the information...

Math
You need to use direct integration to evaluate the general solution to the differential equation: `int (8t^3 + 5)dt = int 8t^3 dt + int 5dt` `int (8t^3 + 5)dt = 8t^4/4 + 5t + c` `int (8t^3 + 5)dt =...

Math
You need to use direct integration to evaluate the general solution to the differential equation: `int (10s  12s^3)ds = int 10s ds  int 12s^3 ds` `int (10s  12s^3)ds = (10/2)s^2  (12/4)s^4 +...

Math
`f''(x)=2` `f'(x)=int2dx` `f'(x)=2x+C_1` Now let's find constant C_1 , given f'(2)=5 `5=2(2)+C_1` `5=4+C_1` `C_1=1` `:.f'(x)=2x+1` `f(x)=int(2x+1)dx` `f(x)=(2x^2)/2+x+C_2` `f(x)=x^2+x+C_2` Now...

Math
`f''(x)=x^2` `f'(x)=intx^2dx` `f'(x)=x^3/3+C_1` Now let's find C_1 , given f'(0)=8 `8=0^3/3+C_1` `C_1=8` `:.f'(x)=x^3/3+8` `f(x)=int(x^3/3+8)dx` `f(x)=(1/3)((x^(3+1))/(3+1))+8x+C_2`...

Math
`int(sqrt(x)+1/(2sqrt(x)))dx` apply the sum rule `=intsqrt(x)dx+int1/(2sqrt(x))dx` Now, `intsqrt(x)dx` apply the power rule `=x^(1/2+1)/(1/2+1)` `=2/3x^(3/2)` `int1/(2sqrt(x))dx`...

Math
You need to find the indefinite integral, hence you need to use the following formula: `int x^n dx = (x^(n+1))/(n+1) + c` Replacing `2/3` for n yields: `int x^(2/3) dx = (x^(2/3+1))/(2/3+1) + c`...

Math
You need to evaluate the indefinite integral, such that: `int (root(4)(x^3) + 1) dx = int root(4)(x^3) dx + int dx` You may use the following formula `int x^n dx = (x^(n+1))/(n+1) + c` `int...

Math
You need to find the indefinite integral, hence you need to use the following formula: `int 1/(x^n) dx = (x^(n+1))/(1n)` Replacing 5 for n yields: `int 1/(x^5) dx = (x^(5+1))/(15)` `int 1/(x^5)...

Math
You need to find the indefinite integral, hence you need to use the following formula: `int 1/(x^n) dx = (x^(n+1))/(1n)` Replacing 7 for n yields: `int 3/(x^7) dx = (3x^(7+1))/(17)` `int...

Math
`int((x+6)/sqrt(x))dx` apply the sum rule `=int(x/sqrt(x))dx+int(6/sqrt(x))dx` Now, `int(x/sqrt(x))dx=intsqrt(x)dx` apply the power rule `=x^(1/2+1)/(1/2+1)` `=2/3x^(3/2)`...

Math
`int((x^43x^2+5)/x^4)dx` apply the sum rule `=intx^4/x^4dxint(3x^2)/x^4dx+int5/x^4dx` `=int1dxint3x^2dx+int5x^4dx` `=x3(x^(2+1)/(2+1))+5(x^(4+1)/(4+1))` `=x+3x^1+5(x^3)/(3)`...

Math
You need to evaluate the indefinite integral, hence, you need to open the brackets such that: `(x + 1)(3x  2) = 3x^2  2x + 3x  2 = 3x^2 + x  2` `int(x + 1)(3x  2)dx = int (3x^2 + x  2)dx` You...

Math
`int(4t^2+3)^2dt` `=int(16t^4+24t^2+9)dt` apply the sum rule and power rule, `=int16t^4dt+int24t^2dt+int9dt` `=16(t^(4+1)/(4+1))+24(t^(2+1)/(2+1))+9t` `=(16t^5)/5+8t^3+9t+C` C is constant

Math
You need to evaluate the indefinite integral, hence, you need to split the integral, such that: `int (5cos x + 4sin x) dx = int 5cos x*dx + int 4sin x*dx` `int 5cos x*dx = 5sin x + c` `int 4sin...

Math
You need to evaluate the indefinite integral, hence, you need to split the integral, such that: `int (t^2  cos t) dt = int t^2 dt  int cos t dt` You need to use the following formula `int t^n dt...

Math
You need to evaluate the indefinite integral, hence, you need to split the integral, such that: `int (x + 7) dx = int x dx  int 7 dx` You need to use the following formula `int x^n dx =...

Math
You need to evaluate the indefinite integral, hence, you need to split the integral, such that: `int (13  x) dx = int 13 dx  int x dx` You need to use the following formula `int x^n dx =...

Math
You need to evaluate the indefinite integral, hence, you need to split the integral, such that: `int (x^5 + 1) dx = int x^5 dx + int dx` You need to use the following formula` int x^n dx =...

Math
You need to evaluate the indefinite integral, hence, you need to split the integral, such that: `int (8x^3  9x^2 + 4) dx = int 8x^3 dx  int 9x^2 dx + int 4dx` You need to use the following...

Math
You need to evaluate the indefinite integral, hence, you need to split the integral, such that: `int (x^(3/2) + 2x + 1) dx = int x^(3/2) dx + int 2x dx + int dx` You need to use the following...