Recently Active Calculus Questions

\hspace{-16}$Period of the function $\mathbf{f(x)},$ Where \\\\ $\mathbf{f(x) = \cos^7 (x)+cos^{7}\left(x+\frac{2\pi}{3}\right)+cos^{7}\left(x+\frac{4\pi}{3}\right)}$ ...

\hspace{-16}$If $\mathbf{I_{1}=\int_{0}^{1}(1-x^4)^7dx}$ and $\mathbf{I_{2}=\int_{0}^{1}(1-x^4)^6dx}$\\\\\\ Then $\mathbf{\frac{29}{4}.\left(\frac{I_{1}}{I_{2}}\right)=}$ ...

\hspace{-16}$For $\mathbf{a\;,b>0}$. Evaluate the Integral\\\\ $\mathbf{\int_{0}^{\infty}\frac{e^{ax}-e^{bx}}{x.(e^{ax}+1).(e^{bx}+1)}dx}$ ...

\hspace{-16}\mathbf{\int\frac{1}{(x^2+a^2).(x^2+b^2).(x^2+c^2)}dx} ...

\hspace{-16}\mathbf{\int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{\sqrt{3-4x-4x^2}}{4x^2+4x+5}dx} ...

What to do and what not to do in the last month before IIT-JEE http://www.youtube.com/watch?v=Z5-AfiJxjOc&context=C4acfebbADvjVQa1PpcFMQX-FKsRHXxJDA7RTt_k4sN1NL6Iapsuo= ...

\hspace{-16}\mathbf{\int_{0}^{\pi}\ln(3+\cos x)dx} ...

\hspace{-16}$If $\mathbf{f(0)=1}$ and $\mathbf{f(1)=2}$ and $\mathbf{f(x)=\frac{f(x+1)+f(x+2)}{2}}$\\\\\\ Then $\mathbf{f(2012)=}$ f(x) = 1/3.[4-(-2)x] ...

\hspace{-16}$If $\mathbf{I_{1}=\int_{0}^{1}\frac{\ln(1+x)}{x}dx}$ and $\mathbf{I_{2}=\int_{0}^{1}\frac{\ln(1-x)}{x}dx}$\\\\\\ and $\mathbf{\left(\frac{1}{I_{1}}+\frac{1}{I_{2}}\right).\pi^2=n}$\\\\\\ Then value of $\mathbf{n} ...

\dpi{120} \hspace{-16}$If $\mathbf{I_{1}=\int_{0}^{\infty}(1+x^2)^{-2012}dx}$ and $\mathbf{I_{2}=\int_{0}^{\infty}(1+x^2)^{-2011}dx}$\\\\\\ Then find value of $\mathbf{\frac{I_{1}}{I_{2}}=}$ Ans=(4021)/(4022) ...

\hspace{-16}\mathbf{\int_{\frac{1}{2}}^{2}\frac{\tan^{-1}(x)}{x^2-x+1}dx} ...

*Image* *Image* *Image* *Image* *Image* *Image* *Image* please provide solutions. ...

\hspace{-16}\displaystyle\mathbf{\lim_{n\rightarrow \infty}\frac{1}{\bold{\sqrt[2011]{n^{2011}+1}}}+\frac{1}{\bold{\sqrt[2011]{n^{2011}+2}}}+.......+\frac{1}{\bold{\sqrt[2011]{n^{2011}+n}}}} ...

\hspace{-16}$(1)\;\; Find No. of $\mathbf{\mathbb{R}}$eal Roots of $\mathbf{e^{ax}=bx}$\\\\ Where $\mathbf{a,b>0}$\\\\\\ $(2)\;\; $No. of $\mathbf{\mathbb{R}}$eal Roots of the equation $\mathbf{3^x+4^x+5^x = x^2}$\\\\\\ $( ...

Find the smallest value for the function : f(x) = |x-2| + |x+3| + |x-4| ...

\hspace{-16}$If $\mathbf{A=\left\{\frac{2012}{2013}\right\}+\left\{2.\frac{2012}{2013}\right\}+............+\left\{2012.\frac{2012}{2013}\right\}}$\\\\\\\\ and $\mathbf{B=\left\{\frac{2014}{2013}\right\}+\left\{2.\frac{2014}{ ...

\hspace{-16}\mathbf{(1)::\int\frac{1}{x+\sqrt{x^2+x+1}}dx}\\\\\\ \mathbf{(2)::\int\frac{x^{2001}}{(1+x^2)^{1002}}dx}\\\\\\ \mathbf{(3)::\int\frac{x^3-x}{x^6+4x^4+4x^2+1}dx} ...

Find the points where f(x) is not differentiable. f(x)=|x-1|3 +|x-2|5+|x-3| ...

Compute: *Image* Where *Image* . ...

\mathbf{\lim_{n\rightarrow \infty}\left(2012.\bold{\sqrt[n]{2010}}-2011\right)^n} ...

\hspace{-16}(1)\;\;\int\frac{25^x}{15^x-9^x}dx\\\\ (2)\;\;\int\frac{1}{1+x^6}dx\\\\ (3)\;\;\int\frac{\sqrt{1-x^2}-x}{x^3-x^2-x+1-\sqrt{1-x^2}+x\sqrt{1-x^2}}dx ...

\hspace{-16}\mathbf{\int\frac{xdx}{\sqrt{1-x^2.(x+\sqrt{1-x^2})^8}}dx}$ ...

\hspace{-16}$Prove that $\mathbf{f(x)=\int_{0}^{\ln (x)}e^t.\sin (\pi.t^2)dt}$. Then prove that \\\\ $\mathbf{\frac{d^2}{dx^2}(f(x))}$ has at least $\mathbf{2}$ positive Real Roots. ...

\hspace{-16}$Find solution of diff. equation $\mathbf{\int_{0}^{\frac{dy}{dx}}\frac{\cos z}{16+9\sin^2 z}dz=\frac{1}{12}\tan^{-1}(x)}$ ans y = x.sin-1(4x/3)-1/4* 9-16x^2 +C ...

Find the limiting value of { (√2 + 1)n} x (-1)[(√2 + 1)n] where n →∞ {} →fractional part [.] →G.I.F. ...

1) \lim_{x\rightarrow \frac{\pi }{2}} sec^{-1}(sin x) 2) \lim_{x\rightarrow 0}cos\frac{1}{x} 3) \lim_{x\rightarrow 0}(1+sinx)^{1/x^{2}} 4) Let f( x+y/2 ) = f(x)+f(y)/2 for all real x and y.If f'(0) = -1 and f(0) = 1,find f(2) ...

Who can do this in min 4 different ways \lim_{x\rightarrow 0}\frac{tan^{-1}x-sin^{-1}x}{sin^{3}x} ...

f(x) is a real valued function satisfying f(x-y)=f(x)f(y)-f(a-x)f(a+x) f(0)=1 then f(2a-x) is a) f(-x) b) f(x) c(-f(x) d f(a) +f(a-x) ...

f(x) is an invertible function and f(x)=xsin x ; g(x)=f -1(x). find the area bounded by y=f(x) and y=g(x). ...

How do we integrate x2/(a+bx)2.????(indefinite) ...