Recently Active Calculus Questions

\int (cos^3x + cos^5x)/ (sin^2x + sin^4x) dx ...

1. Show that x^2>(1+x)[ln(1+x)]^2 \text{ } \forall\text{ } x>0 2. Let a+b=4, where a<2 and let g(x) be a differentiable function. If dg/dx >0 for all x, prove that \int_{0}^{a}{g(x)dx}+\int_{0}^{b}{g(x)dx} increas ...

*Image* ...

$Solve Diff. equations $\frac{dy}{dx}=\frac{3x+x^2y}{y+x^2y} ...

Find \lim_{n\rightarrow \infty}\int_{0}^{1}{}x^{2}e^{-(\frac{x^2}{n^2})}dx ...

\large \dpi{120} \mathbf{\lim_{n\rightarrow \infty} \frac{\binom{n}{2}}{\binom{7n}{8}}.n^6=} ...

\dpi{120} \hspace{-16}(1)::\; \mathbf{\lim_{n\rightarrow \infty}\frac{\sqrt{n}-\sqrt{n-1}+\sqrt{n-2}-...........+(-1)^n.\sqrt{1}}{\sqrt{n}}} ...

\hspace{-16}(1)::\mathbf{\int e^{\sin x}\left(\frac{x.\cos^3 x-\sin x}{\cos^2 x}\right)dx}\\\\\\ (2)::\mathbf{\int\frac{\sin x.\cos x}{\sin x+\cos x}dx}$ ...

1) If c be a positive constant and | f(y) - f(x)| ≤ c(y-x)2, then for all real x and y,then a)f(x)=0 for all x b)f(x)=x for all x c)f '(x)=0 for all x d)f '(x)=c for all x 2) If P(x) = x(x+1)(x+2)........(x+2004),then for p ...

\hspace{-16}(1)\;\;\int\frac{\sin x+\cos x}{\sin^2 x+\cos ^4 x}dx\\\\\\ (2)\;\; \int\frac{\cos x+x.\sin x}{x^2+\cos^2 x}dx asked in goiit ...

Prove this : \lim _{n\rightarrow \infty } \frac{|\sin \theta|+|\sin 2\theta|+...|\sin n\theta|}{n}>0 I must admit that I've its soln. and I'd not have been able to solve it by myself otherwise. But I liked the sum, so flic ...

Q: Let f : [0,1] → R is a continuous function such that \int_{0}^{1}{f(x) dx=0} . Prove that there is some ' c ' ε (0,1) such that \int_{0}^{c}{f(x) dx} = f(c) ...

Q: (x4y2-y) dx + (x2y4-x) dy = 0 Q: \int \frac{x dx}{log x . sec x} ...

Let f be twice differentiable in [0,2]. show that if f(0)=0,f(1)=2, f(2)=4, then there is an 'x' in (0,2) such that f''(x)=0 ...

Use the mean value theorem to prove that |sinx-siny| ≤|x-y| for all real numbers x,y ...

\int \left(tan^{-1}x \right)^{2}dx ...

Let f(x) be a continuous function such that f(x) does not vanish for all real values of x if *Image* then f(x) (for all x ε R) is (a) an even function (b) an odd function (c) a periodic function (d) None of these Ans: (d) ...

Find all continuous functions f:\mathbb{R}\to [1,\infty) for which there exist an a\in\mathbb{R} and k , a positive integer, such that f(x)f(2x)\cdots f(nx)\le an^k for every real number x and positive integer n. ...

If x,y,z≥0 and x2+y2+z2=3 Find the max for the expression F=xy+xz+yz+ 5/x+y+z ...

pls someone help me..in the graph f(x)=1/x(reciprocl function of x) we observe that x(tends) -- 0+ then f(x) -- +infinity and x--0(minus) then f(x)-- minus infinity and when x tends to plus or minus infinty then f(x) tends to ...

1.10.2 F LIM sqrt(2xsquared+3)/4x+2 as x→∞ is given as does not exist!!!! I dont know why. Somebody help me!!! ...

1. Find the largest term of the following sequence: \text{a) } a_{n}=\frac{n^2}{n^3+200} 2. ABC is an isosceles triangle inscribed in a circle of radius r, AB=AC and h is the altitude from A to BC. If the triangle ABC has per ...

Givn the integral of sqrt (x^2 - a^2) is it possible to find the integration of sqrt (a^2 - x^2) without using further integration ...

$If $\mathbf{\color{red}f(x)=\frac{(x-a).(x-b)}{(c-a)(c-b)}+\frac{(x-b).(x-c)}{(a-b)(a-c)}+\frac{(x-c).(x-a)}{(b-c)(b-a)}}$\\\\\\ Then find $\mathbf{\color{green}\frac{d}{dx}\left(f(x)\right)=}$ ...

\hspace{-16}\bold{(Q):(1):}\; \mathbf{\lim_{x\rightarrow 0}\left(\frac{b^{x+1}-a^{x+1}}{b-a}\right)^{\frac{1}{x}}=}$\\\\\\ $\bold{(Q):(2):}\;$If $\mathbf{f(x)=}$\begin{cases} \mathbf{\displaystyle \frac{x^2-1}{x^3-1}}\;\;, & ...

\hspace{-16}(1):\; \mathbf{\int x\sqrt{1+\sin x}dx}\\\\\\ (2):\; \mathbf{\int_{0}^{\pi}\frac{2+2(x+1)\sin x-(x^2+1)\cos^2 x}{\sin x-x\cos x+1}dx}\\\\\\ (3):\;\mathbf{\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos^2 x}{(e^x+1 ...

\hspace{-15}$(1)\;\;::\;If $\mathbf{f:\mathbb{Z}\rightarrow \mathbb{Z}\;,}$ and $\mathbf{f(x)-f(x-1)=x^3}$ and $\mathbf{f(2)=-1}$\\\\ Then $\mathbf{f(x)=}$\\\\ $\mathbf{Ans:\Leftrightarrow f(x)=\left(\frac{n.(n+1)}{2}\right)^ ...

This is created by me i am quite sure it is correct looking for a rigorous proof Find the value of \lim_{n\rightarrow \infty}cos^{(n)}(t) where cos^{(1)}(x)= cos(x) , cos^{(n)}(x)= cos(cos^{(n-1)}(x)) n is a natural number an ...

Q1. \mathrm{f(x)=x+\int_{0}^{1}(xy^2+x^2y)f(y)dy} ,\texttt{Then f(x) attains a minimum at} \\ \\ (A)x=\frac{9}{8} \ \ \ \(B)x=\frac{-9}{8} \\ \\ (A)x=0 \ \ \ \(D)x=1 Q2. \texttt{Let f(x) be a positive , continuous and differe ...

1. \int \left( \sqrt{x+\sqrt{x^2+a^2}}\right)dx ...