Calculus Questions (384)

MATCH THE FOLLOWING: (A) $\int \frac{e^{2x} - 1}{e^{2x} + 1} dx$ is equal to (B) $\int \frac{1}{(e^x + e^{-x})^2} dx$ is equal to (C) $\int \frac{e^x}{1 + e^x} dx$ is equal to (D) $\int \frac{1}{\sqrt{1 - e^{2x}}} dx$ is equal to

A strictly increasing continuous function $f(x)$ intersects its inverse $f^{-1}(x)$ at $x=\alpha$ and $x=\beta$, $\displaystyle\int_\alpha^\beta (f(x)+f^{-1}(x))\,dx=13$, where $\alpha,\beta\in\mathbb{N}$. Then $|\alpha\beta|$ equals

If $I=\displaystyle\int\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}\,dx$, then $I$ equals

$\displaystyle\int\dfrac{2x^3+3x^2-11x-17}{(x^2-x-6)^2}dx$

If $f(x)=\displaystyle\int x^{1/3}(x^{2/3}+x^{1/3}+1)(2x^{2/3}+3x^{1/3}+6)^3\,dx$, $f(0)=0$, then $\dfrac{16}{11}f(1)$ is equal to

Suppose $f(x)$ is a function satisfying the following conditions:\n(i) $f(0) = 2, f(1) = 1$,\n(ii) $f$ has a minimum value at $x = 5/2$\n(iii) For all $x, f'(x) = \begin{vmatrix} 2ax & 2ax - 1 & 2ax + b + 1 \\ b & b + 1 & -1 \\ 2(ax + b) & 2ax + 2b + 1 & 2ax + b \end{vmatrix}$\n$f(x) = 0$ has

Let $f(x)=\displaystyle\int e^x(x-1)(x-2)\,dx$. Then $f(x)$ decreases in the interval

The intercepts on the x-axis made by tangents to the curve $y=\displaystyle\int_0^x|t|\,dt$, $x\in\mathbb{R}$, which are parallel to $y=2x$, are equal to

Let $f,g,h:\mathbb{R}\to\mathbb{R}$ be differentiable with $f(x)=x^5+x^3+3x+7$, $g(f(x))=x$ and $h(g(g(x)))=x$. Value of $h'(-1)$ is

Which of the following functions is differentiable at $x=0$?

$\displaystyle\int_{-1}^{1}\dfrac{d}{dx}\left(\tan^{-1}\dfrac{1}{x}\right)dx=$

The smaller area (in sq. units) included between the curves $\sqrt{x}+\sqrt{|y|}=1$ and $|x|+|y|=1$ is

If $I=\displaystyle\int\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}\,dx$, then $I$ equals

If $\displaystyle\int (x^2+1)(x+1)^2 e^x\,dx = A(f(x))^2 + C$ and $f(-1) = \frac{2}{e}$, then $2A + f(0)$ equals

The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $\left[0,\dfrac{\pi}{2}\right]$ is

Let $P(x)$ be a quadratic polynomial with $P(1)=-1$. If $\displaystyle\int\frac{P(x)\,dx}{(2x-3)^2(3x-2)^2}=\frac{1}{5}\ln|f(x)|+C$ where $f(x)$ is rational and $\lim_{x\to\infty}f(x)=\frac{4}{3}$, then $|f(1)|$ is

The set of all real values of $x$ where $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable is

The value of the definite integral $\displaystyle\int_{-2}^{2}x^3\ln(1^x+3^x+5^x+15^x)\,dx$

Which of the following functions is differentiable at $x=0$?

Let $f$ be a function defined by $f(x)=\begin{cases}(x-r)^2, & r-1\leq x<r+1\\ 1, & r+1\leq x\leq r+2\end{cases}$ where $r=3k$, $k\in I$. Find $\displaystyle\sqrt{\int_0^{45}f(x)\,dx}$

If $\displaystyle\int\frac{\sec^2 x-2010}{\sin^{2010}x}\,dx=\frac{P(x)}{(\sin x)^{2010}}+C$, then the value of $P\!\left(\dfrac{\pi}{3}\right)$ is

The area bounded by the curve $y = \dfrac{1}{x^2 - 2x + 2}$ and the $x$-axis equals

Let $f(x)=\displaystyle\int e^x(x-1)(x-2)\,dx$. Then $f(x)$ decreases in the interval

The area enclosed between the curves $y=ax^2$ and $x=ay^2$ ($a>0$) is 1 sq. unit. Then the value of $a$ is

Let $I=\displaystyle\int_0^2\!\left[\left|x^2-5x+4\right|+\left[\sin\frac{3\pi}{2}x\right]\right]dx$ (where $[\cdot]$ is GIF). Then $I+\dfrac{2}{3}$ is

If $I=\displaystyle\int\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}\,dx$, then $I$ equals

Let $f,g,h:\mathbb{R}\to\mathbb{R}$ be differentiable with $f(x)=x^5+x^3+3x+7$, $g(f(x))=x$ and $h(g(g(x)))=x$. Value of $h'(-1)$ is

If $f'(x) = \frac{1}{-x + \sqrt{x^2 + 1}}$ and $f(0) = -\frac{1 + \sqrt{2}}{2}$, then the value of $f(5)$ will be :

The set of all real values of $x$ where $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable is

$\lim_{x \to \infty} \sqrt[3]{(x+a)(x+b)(x+c)} - x =$

Define $f: [0, \pi] \to \mathbb{R}$ by $f(x) = \begin{cases} \tan^2 x + \sqrt{2\sin^2 x + 3\sin x + 4 - \sqrt{\sin^2 x + 6\sin x + 2}} & x \neq \pi/2 \\ k & x = \pi/2 \end{cases}$ is continuous at $x = \pi/2$, then $k$ is equal to:

The function $f(x) = [x] + \sqrt{\{x\}}$, where $[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function respectively, is discontinuous at

Let $f(x) = \begin{cases} \frac{\tan^2 [x]}{x^2 - [x]^2} & \text{for } x > 0 \\ 1 & \text{for } x = 0 \\ \sqrt{[x]} \cot [x] & \text{for } x < 0 \end{cases}$ where $[x]$ is the step up function and $\{x\}$ is the fractional part function of $x$, then :

The function $f(x) = \sqrt{1 - \sqrt{1 - x^2}}$

The function, $f(x) = [x] - [[x]]$, where $[ ]$ denotes greatest integer function:

$f$ is a continuous function in $[a, b]$; $g$ is a continuous function in $[b, c]$. A function $h(x)$ is defined as: $h(x) = f(x)$ for $x \in [a, b]$ $= g(x)$ for $x \in [b, c]$ if $f(b) = g(b)$, then

Which of the following limits vanish?

$\lim_{x \to c} f(x)$ does not exist when:

Let $f(x) = |x - 1|([x] - [-x])$, then which of the following statement(s) is/are correct. (where $[.]$ denotes greatest integer function.)

The function $f(x) = x^2 \left[x^2 - \frac{1}{x^2}\right], x \neq 0$ is ($[x]$ represents the greatest integer $\leq x$)

If $y = f(x)$ defined parametrically by $x = 2t - |t - 1|$ and $y = 2t^2 + t|t|$, then:

$f(x)$ is defined for $x \geq 0$ and has a continuous derivative. It satisfies $f(0) = 1, f'(0) = 0$ and $(1 + f(x))f''(x) = 1 + x$. The values $f(1)$ can't take is/are:

Consider $f(x) = \int \left(t + \frac{1}{t}\right) dt$ and $g(x) = f'(x)$ for $x \in \left[-3, -\frac{1}{2}\right]$. If P is a point on the curve $y = g(x)$ such that the tangent to this curve at P is parallel to a chord joining the points $\left(\frac{1}{2}, g\left(\frac{1}{2}\right)\right)$ and $(3, g(3))$ of the curve, then the coordinates of the point P

P and Q are two points on a circle of centre C and radius $a$, the angle PCQ being 20 then the radius of the circle inscribed in the triangle CPQ is maximum when

The integral $\int \frac{\sec^{3/2}\theta - \sec^{1/2}\theta}{2 + \tan^2\theta} \tan\theta d\theta$ is:

If $f(x) = \lim_{n \to \infty} \frac{x^n - x^{-n}}{x^n + x^{-n}}$, $0 < x < 1$, $n \in \mathbb{N}$ then $\int (\sin^{-1} x) f'(x) dx$ is equal to:

The derivative of $x^4 + x^{-5}$ is $-\left(4x^{-5} + 5x^{-6}\right)$. So, $$\int \frac{5x^3 + 4x^5}{\left(x^5 + x + 1\right)^2} dx =$$

If the anti-derivative of $\frac{x^3}{\sqrt{4+2x^2}}$ which passes through $(1, 2)$ is $\frac{1}{m}\left(1+2x^2\right)^{1/2}\left(x^2-1\right)+c$. Then:

Let $f(xy) = f(x) \cdot f(y)$, $\forall x > 0, y > 0$ and $f(1 + x) = 1 + x[1 + g(x)]$, where $\lim_{x \to 0} g(x) = 0$, then $$\int \frac{f(x)}{f'(x)} dx$$ is:

If $$\int \frac{dx}{x^2(x^n + 1)^{(n-1)/n}} = -[f(x)]^{1/n} + C$$, then $f(x)$ is