# Recent questions tagged definite-integrals

1
The value of the Integral $I = \displaystyle{}\int_{0}^{\pi/2} x^{2}\sin x dx$ is $(x+2)/2$ $2/(\pi-2)$ $\pi – 2$ $\pi + 2$
2
The value of improper integral $\displaystyle\int_{0}^{1} x\ln x =?$ $1/4$ $0$ $-1/4$ $1$
3
What is the derivative w.r.t $x$ of the function given by $\large \Phi(x)= \displaystyle \int_{0}^{x^2}\sqrt t\:dt$, $2x^2$ $\sqrt x$ $0$ $1$
4
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
5
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 $\underset{t \to \infty}{\lim} A(t)$ equals $0$ $1$ $2$ $4$
6
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 $\alpha$ $[\alpha]$ $1$ $\dfrac{[\alpha] + [\alpha +1]}{2}$
7
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is $\frac{\pi}{6} + \sqrt{3}$ $\frac{\pi}{6} - \sqrt{3}$ $0$ $\frac{1}{2}$
8
The value of the integral $\displaystyle{}\int_{-1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$ $\frac{1}{2}$ $– \frac{1}{2}$ $1$
9
Consider the function $f(x) = \begin{cases} \int_0^x \{5+ \mid 1-y \mid \} dy & \text{ if } x>2 \\ 5x+2 & \text{ if } x \leq 2 \end{cases}$ Then $f$ is not continuous at $x=2$ $f$ is continuous and differentiable everywhere $f$ is continuous everywhere but not differentiable at $x=1$ $f$ is continuous everywhere but not differentiable at $x=2$
10
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 $\sqrt{\pi/3}$ $\pi/\sqrt{3}$ $\sqrt{2 \pi/3}$ $2 \pi / \sqrt{3}$
11
The value of $\displaystyle \lim_{n \to \infty} \left[ (n+1) \int_0^1 x^n \ln(1+x) dx \right]$ is $0$ $\ln 2$ $\ln 3$ $\infty$
12
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 $\log_e \frac{\beta}{\alpha}$ $\log_e \frac{1+ \beta}{1 + \alpha}$ $\log_e \frac{1+\alpha }{1+ \beta}$ $\infty$
13
If $f$ is continuous in $[0,1]$ then $\displaystyle \lim_ {n \to \infty} \sum_{j=0}^{[n/2]} \frac{1}{n} f \left(\frac{j}{n} \right)$ (where $[y]$ is the largest integer less than or equal to $y$) does not exist exists and is equal to $\frac{1}{2} \int_0^1 f(x) dx$ exists and is equal to $\int_0^1 f(x) dx$ exists and is equal to $\int_0^{1/2} f(x) dx$
14
The area bounded by $y=x^2-4$, $y=0$ and $x=4$ is $\frac{64}{3}$ $6$ $\frac{16}{3}$ $\frac{32}{3}$
15
If $f(x) = \begin{vmatrix} 2 \cos ^2 x & \sin 2x & – \sin x \\ \sin 2x & 2 \sin ^2 x & \cos x \\ \sin x & – \cos x & 0 \end{vmatrix},$ then $\int_0^{\frac{\pi}{2}} [ f(x) + f’(x)] dx$ is $\pi$ $\frac{\pi}{2}$ $0$ $1$
16
$\int \frac{x^3}{\sqrt{1+x^2}}.dx$
17
Given that $\int_{-\infty}^{\infty} e^{-x^2/2} dx = \sqrt{2 \pi}$, what is the value of $\int_{- \infty}^{\infty} \mid x \mid ^{-1/2} e^{- \mid x \mid} dx$? $0$ $\sqrt{\pi}$ $2 \sqrt{\pi}$ $\infty$
1 vote
$\int \limits_0^1 (1 + y^2)^{-1.5} dy=?$
$\int_{-4}^{4}|3-x|dx$ a) 13 b)8 c)25 d)24
$f(x) = \int_{- \infty}^{\infty} f(x) e^{2\pi x} dx$ solve f(x)