
Math
The integral `int 1/(1+sqrt x) dx` has to be derived. `int 1/(1+sqrt x) dx` Let `x = u^2` `(dx)/(du) = 2u` `dx = 2u*du ` `int 1/(1+sqrt x) dx` = `int 1/(1+u) * 2u du` Let `1 + u = y => dy = du`...

Math
The definite integral `int_0^8 (1+x^2)/(x^3 + 3x) dx` has to be determined. First determine the integral `int (1+x^2)/(x^3 + 3x) dx` and find the value of the same between x = 0 and x = 8 `int...

Math
The yintercept of the tangent to the curve y = e^x*sin x at the point where x = 0 has to be determined. For a curve y = f(x), the slope of the tangent to the curve at a point `(y_0, x_0)` is...

Math
To find the gradient or slope of the tangent line at a given point on the curve, we need to take the derivative of the equation of that curve, which gives the slope of the tangent line at any...

Math
`f(x)=ax^3+6x^2+bx+4` Differentiating with respect to x, `f'(x)=3ax^2+12x+b` Differentiating again with respect to x, `f''(x)=6ax+12` f has a relative minimum at x=1 and a relative minimum at...

Math
The surface area of a closed cylinder is 100 cm^2. The material used to build this cylinder is used to create another cylinder such that the volume is maximized. Let the radius of the ends of the...

Calculus
Given, velocity of water from the pipe, `v = 1064p^(1/2)` Differentiate both sides with respect to t, `(dv)/(dt)=1064*1/2*p^(1/2)*(dp)/(dt)` `=532* p^(1/2)*(dp)/(dt)` Put the given values:...

Calculus
A rectangular sheet of perimeter 27 cm and dimensions x cm by y cm is to be rolled into a cylinder. What values of x and y give the largest volume? Perimeter of rectangularsheet= 27 cm But...

Calculus
Let x be the distance of the man from the lamp post and y be the distance of the tip of his shadow from the lamp post (please refer to the attached image). Triangles `Delta SBL` and `DeltaHFL` are...

Calculus
Let h be the length (in inches) of the cutout portion of the square corners. Eventually, that would be the height of the cardboard box. Both sides of the cardboard would reduce by 2h, owing to the...

Calculus
The company wishes to manufacture a box with a volume of 44 cubic feet that is open on top and is twice as long as it is wide. Let the length of the box that uses minimum amount of material be L....

Calculus
Given equation of the curve is: `x^4y^4=16` Use implicit differentiation to solve for dy/dx: `d/(dx)(x^4y^4)=d/(dx)(16)` `rArr x^4*4y^3(dy)/(dx)+4x^3y^4=0` `rArr...

Calculus
A. Area=`int_(x=1)^(x=4)ydx` `=int_1^4 1/sqrt(x)dx` `={2sqrt(x)}_1^4` `=2(21)=2` `` Area is 2 square units. B. area=`int_1^3(x^23x+2)dx` `=(x^3/33x^2/2+2x)_1^3` `=(927/2+6)(1/33/2+2)`...

Calculus
You need to evaluate the partial derivative `f_x` differentiate the given function with respect to x, considering y as constant, such that: `f_x = (del f(x,y))/(del x)` `f_x = 1/(sqrt(1 ...

Math
a) Using geometry, the object we wish to find the surface area of is a cone with no base and no 'nose' (a 'conical fustrum'). The formula for the surface area of this cone with no base and no...

Math
We are given the function `y = 3lnx  3x^2` and asked to fund the number of extreme points. An extreme point is an absolute maximum or absolute minimum, absolute meaning that it is finite in...

Calculus
`f(x;y)=x/y` it's defined `D=<<AA x in RR >> xx<< AA y in RR  y!=0 >>` We se function is not limited for, x wit a finte value and y too much close to zero on the left...

Math
The antiderivative of a function is equivalent of the integral of that function. Therefore the antiderivative of `(x^2+2x+2)^(1/2)` with respect to `x` is equivalent to `int (x^2+2x+2)^(1/2)dx`...

Math
The formula for the volume of a sphere is `V = 4/3pir^3` Think of this as adding two hemispheres together, where each of those hemispheres is the sum of many many circle slices/discs from `x=0` ro...

Calculus
1) a) `y = 3/x` , `x>0` The area under the graph is given by `int_0^infty 3/x dx = 3log(x)_0^infty = 3[lim_(x> infty)log(x)  lim_(x>0)log(x)] ` `= lim_(x> infty) x` b) `y = 12x`,...

Calculus
We know by defnition of inverse function. `f^(1)(f(x))=x` Let `g=f^(1)` ,then `g(f(x))=x` (i) differentiating (i) with respect to x ,implicitly `g'(f(x))f'(x)=1` `g'(f(x))=1/(f'(x)) ``(ii)`...

Math
`int_3^6 1/10x(6x^215)dx` First, factor out 1/10. `=1/10int_3^6x(6x^215)dx` Then, distribute x to (6x^215)dx. `=1/10int_3^6(6x^315x)dx` Then, integrate each term inside the parenthesis uisng...

Calculus
Hi, djshan, Sorry, but I'm not too sure what you want us to do here. Are we going to graph this? Find the intercepts? Find the zeros? Something else? I would assume we are graphing it. To...

Math
The solid takes the form of a scaled (by ` ``80pi`) bivariate bell (Normal/Gaussian) curve where the variance of the two variables `x` and `y` is `sigma^2 =2` and the correlation `rho=0` . Ignoring...

Math
We have the surface `f(x,y) = 64 + x^2  y^2` and the constraint `x^2+y^2 <=1` , `x` and `y` `in R` This constraint can be rewritten as `x^2 = 1  y^2` and `y^2 <=1` Substituting for...

Math
I think you mean differential of y = x² sinx. First we find the derivative of sin2x. Using double angle formula sin2x= 2sinx.cosx, we get d(sin2x)/dx = d(2sinx cosx)/dx =...

Calculus
let x+1=t ,dx=dt , x^(n)=(t1)^(n)=t^(n)c(n,1)t^(n1)(1)+ c(n,2)t^(n1)(1)x^(2)+..............+c(n,n1)t (1)^(n1)+ (1)^(n) int(x^(n)/(x+1))dx=int (( t^(n)c(n,1)t^(n1)(1)+...

Calculus
`int_0^1 x^n/(n+1)=x^(n+1)/(n+1)^2_0^1=1/(n+1)^2` So your integral is equal to `1/(n+1)^2` which is less than `1/(n+1)` because n is a natural number. I'm sorry because the above formula is hard...

Calculus
The integral `int (x*e^(2x))/(2x+1)^2 dx` has to be determined. Use integration by parts. Let `x*e^(2x) = u` `du = 2*x*e^(2x) + e^(2x) dx` =>` du = e^(2x)*(1 + 2x) dx` `dx/(2x+1)^2 = dv`...

Calculus
I'm assuming that when you say you have a graph you mean you have graphical representation (picture) of a function which by itself doesn't mean much (you can't do anything better than guessing)....

Calculus
We have to find the value of lim x>0+ [(x^x)^2)] lim x>0+ [(x^x)^2)] => lim x>0+ [e^(ln ((x^x)^2))] => lim x>0+ [e^(ln ((x^2x))] => lim x>0+ [e^(2x*ln x)] As the...

Calculus
We'll have to differentiate with respect to x, using the product rule: (u*v)' = u'*v + u*v' Let u = cos x and v = ln x y' = (cos x)'*(ln x) + (cos x)*(ln x)' y' = sin x*ln x + (cos x)/x The first...

Calculus
We'll rewrite the given expression, isolating dy: dy = (3x^5 + 6e^2x)dx We'll integrate both sides: Int dy = Int (3x^5 + 6e^2x)dx We'll use the property of integral to be additive: Int dy = Int...

Calculus
Since the function that has to be differentiated is the result of composition of 2 functions, logarithmic and linear functions, we'll apply the chain rule and we'll differentiate with respect to...

Calculus
We'll determine f(0). To get f(0), we'll replace x by 0 in the expression of f(x): f(0) = a*0^4 + b*0^2 +c f(0) = c But f(0) = 2 (from enunciation) => c = 2 Now, we'll calculate f'(x): f'(x) =...

Calculus
Int f(x)dx = Int dx/(x+2)(x+3) We'll apply LeibnizNewton rule to calculate the definite integral. We'll decompose the fraction into partial fractions. 1/(x+2)(x+3) = A/(x+2) + B/(x+3) 1 = A(x+3) +...

Calculus
We'll differentiate the given function with respect to x: df/dx = d/dx {[3(1/x)] / (x1)} df/dx = d/dx [3/(x1)]  d/dx [1/x(x1)] d/dx [3/(x1)] = [(x1)*d/dx(3)  3*d/dx(x1)]/(x1)^2 d/dx...

Calculus
We'll have to differentiate the given function y with respect to x. y = sin (cos x) We'll differentiate both sides: dy = [sin (cos x)]'dx We'll differentiate using chain rule, since the given...

Calculus
If we'll replace x by 0, we'll get the indetermination "0/0" type. Therefore, we can use L'Hospital's rule, to determine the limit of the quotient of derivatives. lim f(x)/g(x) = lim f'(x)/g'(x)...

Calculus
We'll have to apply product rule and chain rule to determine the 1st derivative of the given function: y' = (2x)'*ln(2x)*sin(2x) + (2x)*[ln(2x)]'*sin(2x) + (2x)*ln(2x)*[sin(2x)]' y' =...

Calculus
The dimensions of the rectangle, at the time t, are x and y cm. We'll compute the area of the rectangle, at the time t: A = x*y cm^2 (1) We'll have to determine dA/dt if we want to know how fast...

Calculus
Let f(x) = x^x If we'll put x = 1 => f(1) = 1^1 =1 We'll rewrite the function whose limit has to be found out: lim (f(x)  1)/(x  1) By definition, the derivative of a function f(x), at the...

Calculus
The function f(x) = x^(sin x) Let y = f(x) = x^(sin x) Take the natural log of both the sides ln y = ln [ x^(sin x)] => ln y = sin x * ln x Differentiate both the sides with respect to x =>...

Calculus
To determine the primitive function y, we'll have to compute the indefinite integral of the function. If dy/dx = 1/sqrt(36x^2) => dy = dx/sqrt[(6)^2  x^2] We'll integrate both sides: Int dy =...

Calculus
To determine the primitive function f(x), we must calculate the indefinite integral of f'(x). We'll apply substitution technique, replacing ln(2x+3) by t. ln(2x+3) = t We'll differentiate both...

Calculus
We'll write the denominator as the result of expanding the square: x^2 + 4x + 4 = (x+2)^2 We'll rewrite the integral: Int f(x)dx = Int dx/(x+2)^2 We'll use the techinque of changing the variable....

Calculus
I'll solve this problem, considering that the term "182"i s 1*2 (since the "*" symbol and the digit 8 are on the same key and the rest of the terms are written accordingly). The Stolz Cesaro...

Calculus
To determine the antiderivative of a given function, we'll have to calculate the indefinite integral of that function. We'll apply integration by parts. First, we'll recall the formula: Int udv =...

Calculus
We'll manage the right side of the expression and we'll differentiate tan(x/2). [tan(x/2)]' = 1/2*[cos(x/2)]^2 We'll rewrite the expression: (1+sinx)/(1+cosx)=1/2*[cos(x/2)]^2 + tan(x/2) We'll...

Calculus
We have to find lim x> inf. [(3x^24x+1)/(8x^2+5)] substituting x = inf., gives the indeterminate form inf./inf., we can use l'Hopital's rule and substitute the numerator and denominator with...