\(∫\frac {xdx}{(x-1)(x-2)}\ equals\)
For all real values of \(x\), the minimum value of \(\frac{1-x+x^{2}}{1+x+x^{2}}\) is
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is \(\frac{8}{27}\) of the volume of the sphere.
\(\frac{sin^{-1}\sqrt{x}-cos^{-1}\sqrt{x}}{sin^{-1}\sqrt{x}+cos^{-1}\sqrt{x}}\),\(x∈[0,1]\)
\(∫\frac {dx}{x(x^2+1)} \ equals \)
The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\)(x3+xcosx+tan5x+1)dx is
Show that \(\int_{0}^{a}\)ƒ(x)g(x)dx=2\(\int_{0}^{a}\)ƒ(x)dx,if f and g are defined as ƒ(x)=ƒ(a-x)and g(x)+g(a-x)=4
\(\int \sqrt{1+x^2}dx\) is equal to
The maximum value of\( [x(x-1)+1]^{\frac{1}{3}},0≤x≤1\) is
Find the maximum value of 2x3−24x+107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Find the intervals in which the function f given by f(x)=x3+\(\frac{1}{x^3}\),x≠0 is (i) increasing (ii) decreasing
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is \(tan^{-1}\sqrt{2}.\)
Show that the right circular cone of least curved surface and given volume has an altitude equal to\(\sqrt{2}\) time the radius of the base.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i)f(x)=x3,x∈[-2,2] (ii) f(x)=sin x+cos x,x∈[0,π] (iii) f(x)=4x-1/2x2,x∈[-2,\(\frac{9}{2}\)] (iv) f(x)=(x-1)2+3,x∈[-3,1]
Prove that the following functions do not have maxima or minima: (i) f(x) = ex (ii) g(x) = logx (iii) h(x) = x3 + x2+x+1
Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = (2x − 1)2 + 3 (ii) f(x) = 9x2+12x+2 (iii) f(x) = −(x − 1)2+ 10 (iv) g(x) = x3 +1