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Recent questions tagged definite-integrals

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TIFR ECE 2021 | Question: 7
Consider the function \[f(y)=\int_{1}^{y} \frac{1}{1+x^{2}} d x-\log _{e}(1+y)\] where $\log _{e}(x)$ denotes the natural logarithm of $x$. Which of the following is true: The function $f(y)$ ... $y \geq 1$. The derivative of function $f(y)$ does not exist at $y=1$.
Consider the function\[f(y)=\int_{1}^{y} \frac{1}{1+x^{2}} d x-\log _{e}(1+y)\]where $\log _{e}(x)$ denotes the natural logarithm of $x$.Which of the following is true:Th...
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TIFR ECE 2016 | Question: 1
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible? $\int_{1}^{\infty} x f(x) \mathrm{d} x=\infty$ ... $\int_{1}^{\infty} \frac{f(x)}{1+\ln x} \mathrm{~d} x=\infty$ None of the above
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible?$\int_{1}^{\infty} ...
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GATE ECE 2020 | Question: 51
For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.
For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.
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GATE ECE 2019 | Question: 19
The value of the integral $ \displaystyle{}\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.
The value of the integral $ \displaystyle{}\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.
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GATE ECE 2016 Set 3 | Question: 4
The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______
The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______
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GATE ECE 2016 Set 1 | Question: 26
The integral $\frac{1}{2\pi} \iint_D(x+y+10) \,dx\,dy$, where $D$ denotes the disc: $x^2+y^2\leq 4$,evaluates to _________
The integral $\frac{1}{2\pi} \iint_D(x+y+10) \,dx\,dy$, where $D$ denotes the disc: $x^2+y^2\leq 4$,evaluates to _________
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GATE ECE 2015 Set 2 | Question: 27
The value of the integral $\int_{-\infty}^{\infty} 12\cos(2\pi t) \dfrac{\sin(4\pi t)}{4 \pi t}dt$ is _________.
The value of the integral $\int_{-\infty}^{\infty} 12\cos(2\pi t) \dfrac{\sin(4\pi t)}{4 \pi t}dt$ is _________.
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GATE ECE 2015 Set 1 | Question: 29
The maximum area (in square units) of a rectangle whose vertices lie on the eclipse $x^2+4y^2=1$ is __________.
The maximum area (in square units) of a rectangle whose vertices lie on the eclipse $x^2+4y^2=1$ is __________.
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GATE ECE 2014 Set 2 | Question: 45
The value of the integral $\int_{-\infty }^{\infty } \text{sinc}^{2}(5t) \: dt$ is _______.
The value of the integral $\int_{-\infty }^{\infty } \text{sinc}^{2}(5t) \: dt$ is _______.
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GATE ECE 2017 Set 2 | Question: 26
The values of the integrals $\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dy \right )dx$ and $\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dx \right )dy$ are same and equal to $0.5$ same and equal to $-0.5$ $0.5$ and $-0.5$, respectively $-0.5$ and $0.5$, respectively
The values of the integrals $$\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dy \right )dx$$ and $$\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dx \right )dy$$ ...
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