Property 4

Q.

What is the integration of $0$?

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. 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. $\text{Statement-I}:$ The value of the integral $\underset{\pi /6}{\overset{\pi /3}{\int }}\frac{dx}{1+\sqrt{\tan x}}$ is equal to $\frac{\pi }{6}$.
$\text{Statement-II}:\underset{a}{\overset{b}{\int }}f\left(x\right)d\left(x\right)=\underset{a}{\overset{b}{\int }}f(a+b-x)d\left(x\right)$.

1. Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
2. Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.
3. Statement-I is true; Statement-II is false.
4. Statement-I is false; Statement-II is true.

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

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

Q. The value of $\underset{0}{\overset{2\pi }{\int }}\left[\sin 2x\right(1+\cos 3x\left)\right]dx$, where $\left[t\right]$ denotes the greatest integer function, is :

1. $2\pi$
2. $-2\pi$
3. $\pi$
4. $-\pi$

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

Q. ${\int }_0^{\pi /2}\log \left(\sin x\right)dx=$

1. $-\left(\frac{\pi }{2}\right)\log 2$
2. $\pi \log \frac{1}{2}$
3. $-\pi \log \frac{1}{2}$
4. $\frac{1}{2}\log 2$

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. $-\frac{3}{2}\le g\left(2\right)<\frac{1}{2}$
2. $0\le g\left(2\right)<2$
3. $\frac{3}{2}<g\left(2\right)\le \frac{5}{2}$
4. $2<g\left(2\right)<4$

Q. The value of ${\int }_2^8\frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}}\mathrm{dx}$ is

Q. The value of the integral
$\underset{-\pi /2}{\overset{\pi /2}{\int }}{\sin }^{4}x(1+\log \left(\frac{2+\sin x}{2-\sin x}\right))dx$ is :

1. $0$
2. $\frac{3}{4}$
3. $\frac{3}{8}\pi$
4. $\frac{3}{16}\pi$

Q. The value of $\underset{0}{\overset{2\pi }{\int }}\frac{x{\sin }^{8}x}{{\sin }^{8}x+{\cos }^{8}x}dx$ is equal to :

1. $2\pi$
2. $4\pi$
3. $2{\pi }^2$
4. ${\pi }^2$

Q. Let $f:\left(0,1\right]\to R$ be a continuous function such that ${\int }_0^{\pi }f\left(\sin x\right)\mathrm{dx}=2018,\mathrm{then}{\int }_0^{\pi }xf\left(\sin x\right)\mathrm{dx}$ is equal to

1. $1009\pi$
2. $1008\pi$
3. $2017\pi$
4. $2016\pi$

Q. If $f\left(x\right)=\int (\cot \frac{x}{2}-\tan \frac{x}{2})dx,$ where $f\left(\frac{\pi }{2}\right)=0,$ then which of the following statements is (are) CORRECT ?

$($Note : $\text{sgn}\left(y\right)$ denotes the signum function of $y.)$

1. $\underset{0}{\overset{\pi }{\int }}f\left(x\right)dx=-2\pi \ln 2$
2. $\underset{0}{\overset{\pi }{\int }}f\left(x\right)dx=-\pi \ln 2$
3. $\text{sgn}\left(f\left(\frac{2\pi }{3}\right)\right)=-1$
4. $\text{sgn}\left(f\left(\frac{2\pi }{3}\right)\right)=1$

Q. The value of $\underset{0}{\overset{2\pi }{\int }}\frac{x{\sin }^{8}x}{{\sin }^{8}x+{\cos }^{8}x}dx$ is equal to :

1. $2\pi$
2. $4\pi$
3. $2{\pi }^2$
4. ${\pi }^2$

Q. If the value of $\stackrel{\pi /2}{\underset{-\pi /2}{\int }}\frac{x^2}{1+\tan x+\sqrt{1+{\tan }^{2}x}}dx$ is $\frac{{\pi }^3}{a}$. Then $a=$

Q. $If{I}_1={\int }_0^1{2}^{x^2}dx,I_2={\int }_0^1{2}^{x^3}dx,I_3={\int }_1^2{2}^{x^2}dxand{I}_4={\int }_1^{2}2x^{3}dxthen$

1. $I_3>I_4$
2. $I_3=I_4$
3. $I_1>I_2$
4. $I_2>I_1$

Q. Let f(x) be a continuous function of x defined on [0, a] such that f(a - x) = f(x). Then, ${\int }_0^{a}xf\left(x\right)dx$ is equal to

1. $\frac{a}{2}{\int }_0^{a}f\left(x\right)\mathrm{dx}$
2. $a{\int }_0^{a}f\left(x\right)\mathrm{dx}$
3. $\frac{a}{2}{\int }_0^{a}f(a-x)\mathrm{dx}$
4. $a{\int }_0^{a}f(a-x)\mathrm{dx}$

Q. ${\int }_a^{b}si{n}^{2}xdx$ lies in which of the following interval if a, b $ϵ$ R and b > a ?

1. [a, b]
2. [0, a]
3. [0, b ]
4. [0, (b - a )]

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. $-\frac{3}{2}\le g\left(2\right)<\frac{1}{2}$
2. $0\le g\left(2\right)<2$
3. $\frac{3}{2}<g\left(2\right)\le \frac{5}{2}$
4. $2<g\left(2\right)<4$

Q. ${\int }_0^{\pi /2}\frac{x\sin x\cos x}{{\cos }^{4}x+{\sin }^{4}x}dx=$

1. \N
2. $\frac{\pi }{8}$
3. $\frac{{\pi }^2}{8}$
4. $\frac{{\pi }^2}{16}$

Q. If $\underset{-\pi /4}{\overset{3\pi /4}{\int }}\frac{e^{\pi /4}dx}{(e^x+e^{\pi /4})(\sin x+\cos x)}=\lambda \underset{-\pi /2}{\overset{\pi /2}{\int }}\sec xdx,$ then $\lambda$ is equal to

1. $\frac{1}{2}$
2. $\frac{1}{\sqrt{2}}$
3. $\frac{1}{2\sqrt{2}}$
4. $2\sqrt{2}$

Q.

If I is the greatest of the definite integrals
$I_1={\int }_0^1{e}^{-x}co{s}^{2}xdx,I_2={\int }_0^1{e}^{-x^2}co{s}^{2}xdx$
$I_3={\int }_0^1{e}^{-x^2}dx,I_4={\int }_0^1{e}^{\frac{-x^2}{2}}dx,then$

1. $I=I_1$

2. $I=I_2$

3. $I=I_3$

4. $I=I_4$

Q. $If{I}_1={\int }_0^1{2}^{x^2}dx,I_2={\int }_0^1{2}^{x^3}dx,I_3={\int }_1^2{2}^{x^2}dxand{I}_4={\int }_1^{2}2x^{3}dxthen$

1. $I_3>I_4$
2. $I_3=I_4$
3. $I_1>I_2$
4. $I_2>I_1$

Q. If f and g are continuous on [0, a] and satisfy f(x) = f(a - x) and g(x) + g(x - a) = 2, then ${\int }_0^{a}f\left(x\right)g\left(x\right)\mathrm{dx}$is equal to

1. ${\int }_0^{a}f\left(x\right)\mathrm{dx}$
2. ${\int }_0^{a}f(a-x)\mathrm{dx}$
3. ${\int }_0^{a}f(a-x)g(a-x)\mathrm{dx}$
4. ${\int }_0^{a}g\left(x\right)\mathrm{dx}$