GATE Overflow for GATE ESE - Recent questions in Calculus

GATE ES 2026 | Question: 2

Mon, 23 Feb 2026 13:15:46 +0000

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

$-2 \sqrt{2} \sin \dfrac{x}{2}+c$
$-2 \sqrt{2} \cos \dfrac{x}{2}+c$
$-\dfrac{1}{\sqrt{2}} \cos \dfrac{x}{2}+c$
$-\dfrac{1}{\sqrt{2}} \sin \dfrac{x}{2}+c$

GATE ES 2026 | Question: 3

Mon, 23 Feb 2026 13:15:44 +0000

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

$-6$
$-36$
$-26$
$-46$

GATE ES 2026 | Question: 54

Mon, 23 Feb 2026 13:14:21 +0000

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

GATE ES 2025 | Question: 1

Thu, 06 Mar 2025 16:22:57 +0000

Assuming $s>\mid a \mid$; Laplace transform of $f(x)=\cosh a x$ is

$\frac{s}{s^{2}+a^{2}}$
$\frac{a}{s^{2}+a^{2}}$
$\frac{s}{s^{2}-a^{2}}$
$\frac{a}{s^{2}-a^{2}}$

GATE ES 2025 | Question: 3

Thu, 06 Mar 2025 16:22:52 +0000

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 the correct option from the following

$P$ is convergent series; $Q$ is divergent series
$P$ is divergent series; $Q$ is convergent series
Both $P$ and $Q$ are convergent series
Both $P$ and $Q$ are divergent series

GATE ES 2025 | Question: 24

Thu, 06 Mar 2025 16:21:51 +0000

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

GATE ES 2025 | Question: 39

Thu, 06 Mar 2025 16:21:14 +0000

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

GATE ES 2024 | Question: 32

Mon, 19 Feb 2024 10:24:08 +0000

Consider the function $f(x)=\ln (\sin (x))$

Expand $f(x+h)$ using Taylor's series. In this context, the correct statement(s) is/are

Second term in the Taylor's series i.e., the term which includes $h$ is: $h \cdot \ln (\sin (x))$
First term is $\ln (\sin (x))$
Third term in the Taylor's series i.e., the term which includes $h^{2}$ is: $\frac{-h^{2}}{2(\sin (x))^{2}}$
Third term in the Taylor's series i.e., the term which includes $h^{2}$ is: $\frac{2 h^{2}}{(\sin (x))^{2}}$

GATE ES 2024 | Question: 39

Mon, 19 Feb 2024 10:24:07 +0000

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 $x=2$ is ________ $\text{(rounded off to $2$ decimal places)}$

GATE ES 2023 | Question: 3

Tue, 23 May 2023 03:12:48 +0000

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; $\text{Q}$ is divergent series
$\text{P}$ is divergent series; $\text{Q}$ is convergent series
Both $\text{P}$ and $\mathrm{Q}$ are convergent series
Both $\text{P}$ and $\text{Q}$ are divergent series

GATE ES 2023 | Question: 23

Tue, 23 May 2023 03:12:20 +0000

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

GATE ES 2023 | Question: 39

Tue, 23 May 2023 03:12:00 +0000

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

GATE ES 2021 | Question: 2

Tue, 14 Mar 2023 03:44:52 +0000

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

$0$
$1$
$2$
$-1$

GATE ES 2021 | Question: 1

Tue, 14 Mar 2023 03:44:52 +0000

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

$3$
$6$
$9$
$12$

GATE ES 2022 | Question: 24

Fri, 17 Feb 2023 07:05:53 +0000

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

GATE ES 2022 | Question: 42

Fri, 17 Feb 2023 07:05:41 +0000

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.