Recent questions tagged definite-integral

4 votes
1 answer
1
​​​​​Let $f(x)$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ such that\[f(x)=1-f(2-x)\]Which one of the following options is the CORRECT value of ...
5 votes
2 answers
2
The value of the definite integral \[\int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1}\left(4 x^{2} y-z^{3}\right) \mathrm{d} z \mathrm{~d} y \mathrm{~d} x\]is _________. (Rounde...
1 votes
1 answer
3
1 votes
2 answers
6
2 votes
1 answer
7
The integral $$\int _0^{\frac{\pi}{2}} \frac{\sin^{50} x}{\sin^{50}x +\cos^{50}x} dx$$ equals$\frac{3 \pi}{4}$$\frac{\pi}{3}$$\frac{\pi}{4}$none of these
0 votes
1 answer
8
If $A(t)$ is the area of the region bounded by the curve $y=e^{-\mid x \mid}$ and the portion of the $x$-axis between $-t$ and $t$, then $\underset{t \to \infty}{\lim} A(...
3 votes
1 answer
9
For real $\alpha$, the value of $\int_{\alpha}^{\alpha+1} [x]dx$, where $[x]$ denotes the largest integer less than or equal to $x$, is$\alpha$$[\alpha]$$1$$\dfrac{[\alph...
1 votes
1 answer
10
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is$\frac{\pi}{6} + \sqrt{3}$$\frac{\pi}{6} - \sqrt{3}$$0$$\frac{1}{2}$
0 votes
1 answer
11
The value of the integral $\displaystyle{}\int_{-1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$$\frac{1}{2}$$ – \frac{1}{2}$$1$
0 votes
1 answer
13
Given that $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$, the value of $$ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+xy+y^2)} dxdy$$ is$\sqrt{\pi/3}$$...
0 votes
2 answers
14
The value of $$\displaystyle \lim_{n \to \infty} \left[ (n+1) \int_0^1 x^n \ln(1+x) dx \right]$$ is$0$$\ln 2$$\ln 3$$\infty$
0 votes
0 answers
15
Let $0 < \alpha < \beta < 1$. Then $$ \Sigma_{k=1}^{\infty} \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{1}{1+x} dx$$ is equal to$\log_e \frac{\beta}{\alpha}$$\log_e \frac{1+ ...
1 votes
1 answer
17
The area bounded by $y=x^2-4$, $y=0$ and $x=4$ is$\frac{64}{3}$$6$$\frac{16}{3}$$\frac{32}{3}$
1 votes
1 answer
18
7 votes
2 answers
19
Consider the integral$$\int^{1}_{0} \frac{x^{300}}{1+x^2+x^3} dx$$What is the value of this integral correct up to two decimal places?$0.00$$0.02$$0.10$$0.33$$1.00$
0 votes
1 answer
20
0 votes
0 answers
21
Given that $\int_{-\infty}^{\infty} e^{-x^2/2} dx = \sqrt{2 \pi}$, what is the value of $\int_{- \infty}^{\infty} \mid x \mid ^{-1/2} e^{- \mid x \mid} dx$?$0$$\sqrt{\pi}...
2 votes
3 answers
23