
Calculus
We have the function y = e^ln(x^x) take the log of both the sides ln y = ln (e^ln(x^x)) => ln y = ln(x^x) => ln y = x*ln x differentiate both the sides (1/y)dy/dx = ln x + x/x dy/dx = y*(ln x...

Calculus
We have to determine the definite integral of y = sin 2x /sqrt(1 + (sin x)^4), x = 0 to x = pi/2 Int [ sin 2x /sqrt(1 + (sin x)^4) dx] let 1 + (sin x)^2 = y dy/dx = 2*sin x* cos x = sin 2x => dy...

Calculus
We have to integrate [1/ (y^2 + 8y + 20) dy] Int [1/( y^2 + 8y + 20) dy] => Int [ 1 / (y^2 + 8y + 16 + 4) dy] => Int [ 1/((y + 4)^2 + 2^2) dy] if u = y + 4 , dy = du => Int [ 1/ ( u^2 +...

Calculus
We have to determine lim x>0 [(2x  sin 2x)/x^3] If we substitute x = 0, we get the indeterminate form 0/0, so we use the l'Hopital's Rule and substitute the numerator and denominator with...

Calculus
We are given that f(x)=1+2x^5/x^2 = 1 + 2x^3. We have to find: lim x >1 [(f(x)  f(1))/(x1)] => lim x >1 [(1+ 2x^3  1  2)/(x1)] => lim x >1 [(2x^3  2)/(x1)]\ => lim x...

Calculus
We need to find the value of lim x>0 [ tan 4x / tan 2x] If we substitute x = 0, we get the indeterminate form 0/0. This allows us to use l'Hopital's rule and substitute the numerator and the...

Calculus
We have to find the value of lim x> 0[ (sin 5x  sin 3x)/x] if we substitute x = 0, we get the form 0/0, which allows us to use the l'Hopital's rule and substitute the numerator and the...

Calculus
We need to find the value of lim x> pi/4 [sin x/(1 2*(sin x)^2)  cos x/2*(cos x)^2  1)  sin 2x / cos x. Substituting x = pi/4, gives us an indeterminate value. lim x> pi/4 [sin x/(1...

Calculus
The critical points are determined by differentiating the function and equating the derivative to 0. It is solved to determine x. f(x) = sin x + cos x f'(x) = cos x  sin x = 0 => cos x = sin...

Calculus
We have to prove that lim x>0 [(a^x  1)/x] = ln a First, if we substitute x = 0, we get the indeterminate form 0/0. This allows the use of l"Hopital's rule and we can substitute the numerator...

Calculus
You want the limit of y=(1cos 2x)/x^2 while x approaches 0. y = (1cos 2x)/x^2 => [1  (1  2*(sin x)^2)]/x^2 => 2*(sin x)^2/x^2 => 2*(sin x / x)^2 lim x> 0 (sin x / x) = 1 Using...

Calculus
We need to determine the integral of (cos x)^7 * sin x. Int [(cos x)^7 * sin x dx] let cos x = u =>  du = sin x dx => Int [ u^7 du] => u^8 / 8 + C substitute u = cos x =>  (cos...

Calculus
We have to determine the value of lim x> 0[(cos x  cos 3x) / x*sin x If we substitute x = 0, we get (1 1) / 0 = 0/0 As this is an indeterminate form we use l'Hopital's rule and replace the...

Calculus
The extreme values of a function occur at the points where the derivative is equal to 0. f(x) = 2x^3 + 3x^2  12x + 5 => f'(x) = 6x^2 + 6x  12 6x^2 + 6x  12 = 0 => x^2 + x  2 = 0 => x^2...

Calculus
We have y=(1+x^2)^3 We have to find dy/dx. We can use the chain rule here. dy/dx = 3(1 + x^2)^2*2x => dy/dx = 6x(1 + x^2)^2 The required result is 6x*(1 + x^2)^2

Calculus
First we need to determine the points of intersection between lnx and ln^2 x ==> ln x = ln^2 x ==> ln^2 x  ln x = 0 ==> lnx ( lnx 1) =0 ==> lnx = 0 ==> x = 1 ==> lnx1 = 0...

Calculus
We have to find the area enclosed between y=x^2  2x + 2 and y = x^2 + 6. First lets find the points of intersection x^2  2x + 2 = x^2 + 6 => 2x^2  2x  4 = 0 => x^2  x  2 = 0 => x^2...

Calculus
The area of the region bounded by the curve y = sqrt (x  1), the y axis and the lines y = 1 and y = 5 is the limited integral of the expression of x in terms of y, between y = 5 and y = 1. y =...

Calculus
We have to verify that lim x>0 [ ln(1+x)/x] = 1. substituting x = 0, we get the indeterminate form 0/0, therefore we can use the l'Hopital's rule and substitute the numerator and denominator...

Calculus
We have the functions f(x) = 3x+ 2 and g(x) = x^2 + 1 u = fog ( x) = f(g(x)) => f(x^2 + 1) => 3(x^2 + 1) + 2 => 3x^2 + 3 + 2 => 3x^2 + 5 v = gof(x) = g(f(x)) => g( 3x + 2) => (3x...

Calculus
We have to find the limit of f(x)=(sin xcos x)/cos 2x for x> 45 degrees. We know that cos 2x = (cos x)^2  (sin x )^2 lim x> 0 [(sin xcos x)/cos 2x] => lim x> 0 [(sin xcos...

Calculus
We have to find the value of (x^2+2x3)/x1 as x approaches from the left. As x approaches from the left x  1 is always negative, so we have x  1 = (1  x) lim x> 1 [ (x^2+2x3)/(1 ...

Calculus
We have dy/dx = 4x^3 + 4x. dy/dx = 4x^3 + 4x => dy = (4x^3 + 4x) dx Integrate both the sides Int [ dy ] = Int [ (4x^3 + 4x) dx ] => y = 4x^4 / 4 + 4*x^2 / 2 => y = x^4 + 2*x^2 + C As the...

Calculus
We first determine the points where the curves y = 8  x^2 and y = x^2, meet. 8  x^2 = x^2 => x^2 = 4 => x = 2 , x = 2 Now we find the integral of 8  x^2  x^2 between the limits x = 2...

Calculus
We have to find the value of lim h>0[[(3+h)^29]/h] lim h>0[[(3+h)^29]/h] => lim h>0[[(3 + h  3)(3 + h + 3)/(3 + h  3)] cancel (3 + h  3) => lim h>0[[(3 + h + 3)]...

Calculus
We have to find Int [1/ (1 + 4x^2) dx]. First substitute u = 2x => du /dx = 2 => du /2 = dx Now Int [1/ (1 + 4x^2) dx] => Int [(1/2)*(1/ (1+u^2) du] => (1/2)*Int [1/ (1 + u^2) du] Now...

Calculus
We have to differentiate f(x) = x*cos 2x f'(x) = x'*cos 2x + x*(cos 2x)' f'(x) = cos 2x + x*(sin 2x)*2 f'(x) = cos 2x  2x*(sin 2x) The required derivative of f(x) = x*cos 2x is f'(x) = cos 2x ...

Calculus
We have to find the value of the definite integral of x^2/sqrt (x^3 + 1) between the limits x = 2 and x = 3. First we determine the indefinite integral and then substitute the values x = 3 and x =...

Calculus
To find the curve we integrate the given dy/dx = 3x^2  2x. Int [ 3x^2  2x dx ] => 3*x^3 / 3  2x^2 / 2 + C => x^3  x^2 + C As the curve passes through (2 , 5) 5 = 2^3  2^2 + C => 5 =...

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
We have to find the value of lim x> 0 [ ln(1+x)/(sinx+sin3x)] substituting x = 0, we get the indeterminate form 0/0. Therefore we can use l'Hopital's Rule and substitute the numerator and...

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...

Calculus
We have to find Int [e^2x * cos 3x dx] Here the best way to solve would be to use integration by parts. Int [u dv] = u*v – Int [v du] take u = e^2x, du = 2*e^2x dx dv = cos 3x dx, v = (1/3)* sin...

Calculus
We have to find the antiderivative of y = x / sqrt ( x^2  9) Let u = x^2  9 => du / dx = 2x => x dx = du/2 Int [ x / sqrt ( x^2  9) dx] => Int [(1/2)*(1/ sqrt u) du] => Int [ (1/2)*...

Calculus
We have to find y' for y = (2x)^(sqrt x) Use natural logariths for both the sides ln y = ln[ (2x)^(sqrt x)] use the property ln a^x = a*ln x => ln y = (sqrt x)*ln ( 2  x) Do implicit...

Calculus
To find the slant asymptote of x^3 / (x + 2)^2 we have to divide x^3 by (x + 2)^2 (x^2 + 4x + 4)  x^3........................................... x  4 ...........................x^3 + 4x^2 + 4x...

Calculus
We have to find the derivative of y = arc sin x/(1x^2). We use the quotient rule here: y' = [(arc sin x)'*(1  x^2)  ( arc sin x)*(1  x^2)']/(1  x^2)^2 => [sqrt(1x^2)*(1  x^2) + 2x*(arc...

Calculus
We have to find the value of lim x> 90[ (1 sin x)/(cos x)^2] substituting x = 90 degrees, we get the indeterminate form 0/0, so we can use l'Hopital's rule and substitute the numerator and...

Calculus
We have to find the derivative of y = (10 + lg (x^10) + e^10x)^10. We use the chain rule to find the derivative of y. y' = 10 * (10 + lg (x^10) + e^10x)^9 * (10 / x + 10*e^10x) => 10 * (10 + lg...

Calculus
We have to find the integral of f'(x)=11e^x/(11+e^x) f'(x)=11e^x/(11+e^x) let 11 + e^x = y e^x dx = dy Int [ 11e^x/(11+e^x) dx] => Int [ 11dy/y] => 11*ln y + C substitute y = 11 + e^x =>...

Calculus
We have the function y = sin x + cos 3x. The derivative of sin x is cos x and the derivative of cos x is sin x. Also, for a function of the form y= f(g(x)), the derivative of y or y' is given by...

Calculus
We have to find the integral of 1/(16x^2+24x+9) 1/(16x^2+24x+9) => 1/(4x+3)^2 let 4x + 3 = u => du/dx = 4 => dx = du/4 Int[ 1/(16x^2+24x+9) dx] => Int [ 1/u^2 du/4] => (1/4) Int [...

Calculus
We have to find the value of lim x> pi[ sin 5x / sin x] We see that substituting x with pi gives us the form 0/0 which is indeterminate. We can use therefore use l'Hopital's rule and use the...

Calculus
We have f(x) = (3x + 1)*e^x We use the product rule to find f'(x) f'(x) = (3x + 1)'*e^x + (3x + 1)*(e^x)' => 3*e^x  (3x +1)e^x => 3*e^x  f(x) f''(x) = 3e^x  f'(x) => 3e^x ...

Calculus
We need to find f(x) given that f'(x) = sin 2x /((sin x)^2  4) let ((sin x)^2  4) = y dy = 2*sin x * cos x dx =>dy = sin 2x dx Int [ sin 2x /((sin x)^2  4) dx] => Int [ 1/y dy] => ln...

Calculus
We have to find lim x> 2 [(x^22x8)/(x^3+8)] using l'Hopital's rule. First we find out if the l'Hopital's Rule can be used here. Substituting x = 2 we get the indeterminate form 0/0;...

Calculus
We have the function: f(x)=12x^4+24x^2+56 f(x) = 12x^4 + 24x^2 + 56 f'(x) = 48x^3 + 48x If f'(x) = 0 => 48x^3 + 48x = 0 => x^3 + 3 = 0 => x( x^2 + 1) = 0 x1 = 0 x2 = sqrt (1) => x2 =...

Calculus
The area bound by the curve y = cosx/(sin^2x4), the x axis and the lines x = 0 and x = pi/2 is the integral of y between the limits x = 0 and x = pi/2. y = cos x/ ((sin x)^2  4) let sin x = y...

Calculus
We have to find the antiderivative of [ cos 2x  (cos x)^2]^1 f(x) = [ cos 2x  (cos x)^2]^1 => f(x) = [(cos x)^2  (sin x)^2  (cos x)^2]^1 => f(x) = (sin x)^2 => f(x) = 1 / (cosec...

Calculus
To find the maximum and minimum values of the function f(x) = 2x^3  4x + 3, we need to find the first derivative. f(x) = 2x^3  4x + 3 f’(x) = 6x^2  4 Equate this to zero and solve for x 6x^2 ...