Property 4

Q. For a function $f\left(x\right)\ge 0\forall xϵ(a,b);a<b\left|{\int }_a^{b}f\right(x\left)dx\right|=5thenfind{\int }_a^b|f\left(x\right)|dx$ ?

1. 3
2. 4
3. 5
4. 6

Q. If $I={\int }_0^1\frac{sinx}{\sqrt{x}}dxandJ={\int }_0^1\frac{cosx}{\sqrt{x}}dx,$ then, which one of the following is true?

1. $I>\frac{2}{3}andJ<2$
2. $I>\frac{2}{3}andJ>2$
3. $I<\frac{2}{3}andJ<2$
4. $I<\frac{2}{3}andJ>2$

Q.

Let $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$ where $\frac{1}{2}\le f\left(t\right)\le 1,tϵ[0,1]$ and $0\le f\left(t\right)\le \frac{1}{2}$ for $tϵ(1,2)$, then [IIT Screening 2000]

1. 
2. 
3. 
4. 

Q. ${\int }_1^5{e}^{(}{x}^2)dx$ lies in which of the following interval ?

1. $(4e,4e^{25})$
2. $(0,4e^{25})$
3. $BothA\&B$
4. $Ca{n}^{\prime }tbedetermined.$

Q. The value of the definite integral ${\int }_0^1\frac{xdx}{x^3+16}$ lies in the interval [a, b]. The smallest such interval is.

1. 

2. [0, 1]

3. 
   

4. None of these

Q. If $f\left(x\right)=\frac{e^x}{1+e^x},I_1={\int }_{f(-a)}^{f\left(a\right)}xg\left\{x\right(1-x\left)\right\}dxand{I}_2={\int }_{f(-a)}^{f\left(a\right)}g\left\{x\right(1-x\left)\right\}dx,$ then the value of $\frac{I_2}{I_1}is$

1. 2
2. 1
3. -1
4. -3

Q. For a function $f\left(x\right)\ge 0\forall xϵ(a,b);a<b\left|{\int }_a^{b}f\right(x\left)dx\right|\le {\int }_a^b|f\left(x\right)|dx$.

1. True
2. False

Q. Let $I_1={\int }_0^{\frac{\pi }{4}}{x}^{2008}(tanx{)}^{2008}dx,I_2={\int }_0^{\frac{\pi }{4}}{x}^{2009}(tanx{)}^{2009}dxand{I}_3={\int }_0^{\frac{\pi }{4}}{x}^{2010}(tanx{)}^{2010}dx$

1. $I_3<I_2<I_1$
2. $I_2<I_3<I_1$
3. $I_1<I_2<I_3$
4. $I_3<I_1<I_2$

Q. $|{\int }_{10}^{19}\frac{sinxdx}{1+x^8}|islessthen$

1. ${10}^{-10}$
2. ${10}^{-11}$
3. ${10}^{-7}$
4. ${10}^{-9}$

Q. If $\underset{0}{\overset{1}{\int }}\frac{{\tan }^{-1}x}{{\cot }^{-1}(1-x+x^2)}dx$ equals $k,$ then the value of $\frac{1}{k}$ is