Recent questions in Calculus

The correct integral of $\int \dfrac{\sin x}{\sqrt{1+\cos x}} d x$ (assuming integral constant as $c$ ) is $\_\_\_\_$ .$-2 \sqrt{2} \sin \dfrac{x}{2}+c$$-2 \sqrt{2} \cos ...

Given $f(x)=3 x^{4}+4 x^{3}-12 x^{2}+6$, the minimum value of $f(x)$ is $\_\_\_\_$.$-6$$-36$$-26$$-46$

The area in the first quadrant bounded by the function $y=(8-x)$ and the coordinate axes is $\_\_\_\_$ square units (answer in integer).

​​​​Assuming $s>\mid a \mid$; Laplace transform of $f(x)=\cosh a x$ is$\frac{s}{s^{2}+a^{2}}$$\frac{a}{s^{2}+a^{2}}$$\frac{s}{s^{2}-a^{2}}$$\frac{a}{s^{2}-a^{2}}$

​​​Consider the following two series$$\begin{array}{l} \mathrm{P}: \sum_{n=1}^{\infty} \frac{1}{n} \\ \mathrm{Q}: \sum_{n=1}^{\infty} \frac{1}{n^{2}} \end{array}$$Choose ...

$\lim _{x \rightarrow 0}\left(\frac{\ln (1+x)}{2 \sin x}\right)$ is $\_\_\_\_\_\_$ (rounded off to two decimal places).

The value of $\int_{0}^{\infty} \frac{\sin 4 x}{\pi x} d x$ is $\_\_\_\_\_\_$ (rounded off to two decimal places)

Consider the function $f(x)=\ln (\sin (x))$Expand $f(x+h)$ using Taylor's series. In this context, the correct statement(s) is/areSecond term in the Taylor's series i.e.,...

Consider the equation for a curve, $y=f(x)=x^{2}+x$.The area enclosed by the curve, the $x$-axis $\text{( $y=0$ line)}$; the vertical lines passing through $x=1$ and $x=2...

Given are two infinite series$\qquad P: \sum \frac{n^{2}+1}{n^{2}}$ $\qquad Q: \sum\left(1+\frac{1}{n}\right)^{-n}$The correct choice is$\text{P}$ is convergent series; $...

The value of $\frac{4}{\pi} \int_{0}^{\pi / 2} \sin ^{2} x d x$ is ___ ____ $\text{(rounded off to two decimal places)}$.

For the function } f(x)=x \sqrt{4-x^{2}}, the maximum value in the range $-2 \leq x \leq 2$ is ____________ $\text { (rounded off to two decimal places). }$

$\lim _{x \rightarrow 0} \frac{x^{2}}{\sin x}=$$0$ $1$$2$$-1$

A tangent is drawn on the curve of the function $y=x^{2}$ at the point $(x, y)=(3,9)$. The slope of the tangent is$3$$6$$9$$12$

$\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}$ is (rounded off to one decimal place)

The area of the region (rounded off to one decimal place) enclosed between the curves $y=x$ and $y=3 \sqrt{x}$ and between the lines $x=0$ and $x=1$ is ________ units.