Property 1

Q. ${\int }_0^1\frac{dx}{[ax+b(1-x){]}^2}=$

1. $\frac{a}{b}$
2. $\frac{b}{a}$
3. $ab$
4. $\frac{1}{ab}$

Q. The value of the integral I = ${\int }_0^{1}x(1-x{)}^{n}dx$ is

1. $\frac{1}{n+1}$
2. $\frac{1}{n+2}$
3. $\frac{1}{n+1}-\frac{1}{n+2}$
4. $\frac{1}{n+1}+\frac{1}{n+2}$

Q.

Which inverse trigonometric function has a range of $\left(0,\mathrm{\pi }\right)$ ?

Q. ${\int }_0^{\frac{\pi }{4}}\frac{sinx+cosx}{9+16sin2x}dx=$

1. $\frac{1}{20}log3$
2. $log3$
3. $\frac{1}{20}log5$
4. $Noneofthese$

Q. ${\int }_0^{\frac{\pi }{2}}\frac{dx}{a^{2}co{s}^{2}x+b^{2}si{n}^{2}x}$ where (a, b >0) is equal to-

1. $\frac{\pi }{2ab}$
2. $\frac{2}{\pi ab}$
3. $\frac{\pi }{2a}$
4. $\frac{\pi }{2b}$

Q. ${\int }_0^{1}si{n}^{-1}\left(\frac{2x}{1+x^2}\right)dx=$

1. $\frac{\pi }{2}-2log\sqrt{2}$
2. $\frac{\pi }{2}+2log\sqrt{2}$
3. $\frac{\pi }{4}-log\sqrt{2}$
4. $\frac{\pi }{4}+log\sqrt{2}$

Q. ${\int }_0^a\frac{x^{4}dx}{(a^2+x^2{)}^4}=$

1. $\frac{1}{16a^3}(\frac{\pi }{4}-\frac{1}{3})$
2. $\frac{1}{16a^3}(\frac{\pi }{4}+\frac{1}{3})$
3. $\frac{1}{16}{a}^3(\frac{\pi }{4}-\frac{1}{3})$
4. $\frac{1}{16}{a}^3(\frac{\pi }{4}+\frac{1}{3})$

Q. If $I={\int }_0^{1}cos\left(2Co{t}^{-1}\sqrt{\frac{1-x}{1+x}}\right)dxthen$

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

Q. ${\int }_0^{log5}\frac{e^x\sqrt{e^x-1}}{e^x-3}dx=$

1. $3+2\pi$
2. $4-\pi$
3. $2+\pi$
4. $none$

Q. The value of the integral ${\int }_0^3\frac{dx}{\sqrt{x+1}+\sqrt{5x+1}}dxis$

1. $\frac{11}{15}$
2. $\frac{14}{15}$
3. $\frac{2}{5}$
4. $1+\frac{1}{2}log\left(\frac{3}{5}\right)$

Q. ${\int }_0^{\frac{\pi }{2}}\frac{cosx}{(1+sinx)(2+sinx)}dx=$

1. $log\frac{4}{3}$
2. $log\frac{1}{3}$
3. $log\frac{3}{4}$
4. $Noneofthese$

Q. If $I={\int }_a^{b}f\left(g\right(x\left)\right).g^{\prime }\left(x\right)dx$ then on substituting g(x) = t where the equation g(x) = t is continuous in the interval [a, b] , I will be equal to -

1. $I={\int }_a^{b}f\left(t\right).dt$
2. $I={\int }_{g\left(a\right)}^{g\left(b\right)}f\left(t\right).dt$
3. $I={\int }_{g\left(b\right)}^{g\left(a\right)}f\left(t\right).dt$
4. None of these

Q. $If{\int }_0^k\frac{dx}{2+8x^2}=\frac{\pi }{16},thenk=$

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

Q. Let $f$ be a differentiable function from $R$ to $R$ such that $|f\left(x\right)-f\left(y\right)|\le 2|x-y{|}^{3/2}$, for all $x,y\in R$. If $f\left(0\right)=1$, then $\underset{0}{\overset{1}{\int }}{f}^2\left(x\right)dx$ is equal to :

1. $0$
2. $1$
3. $\frac{1}{2}$
4. $2$

Q. Let $\frac{d}{dx}F\left(x\right)=\left(\frac{e^{sinx}}{x}\right),x>0.If{\int }_1^4\frac{3}{x}{e}^{sin{x}^3}dx=F\left(k\right)-F\left(1\right)$, then one of the possible values of k is

1. 15
2. 16
3. 63
4. 64

Q. The value of integral ${\int }_{\frac{1}{\pi }}^{\frac{2}{\pi }}\frac{sin\left(\frac{1}{x}\right)}{x^2}dx=$

1. 2
2. -1
3. \N
4. 1

Q. ${\int }_0^{\infty }\frac{log(1+x^2)}{1+x^2}dx=$

1. $\pi log\frac{1}{2}$

2. $\pi log2$
3. $2\pi log\frac{1}{2}$
4. $2\pi log2$

Q. ${\int }_0^1\frac{x}{1+\sqrt{x}}dx=$

1. $2[\frac{5}{6}+log2]$
2. $2[\frac{5}{6}-log2]$
3. $[\frac{5}{6}-log2]$
4. $[\frac{5}{6}+log2]$

Q. ${\int }_0^{\frac{x}{2}}\sqrt{cos\theta }si{n}^3\theta d\theta =$

1. $\frac{20}{21}$
2. $\frac{8}{21}$
3. $\frac{-20}{21}$
4. $\frac{-8}{21}$

Q. ${\int }_0^2\sqrt{\frac{2+x}{2-x}}dx=$

1. $\pi +2$
2. $\pi +\frac{3}{2}$
3. $\pi +1$
4. $Noneofthese$

Q. ${\int }_0^{1}ta{n}^{-1}xdx=$

1. $\frac{\pi }{4}-\frac{1}{2}log2$
2. $\pi -\frac{1}{2}log2$
3. $\frac{\pi }{4}-log2$
4. $\pi -log2$

Q. The value of the integral I = ${\int }_0^{1}x(1-x{)}^{n}dx$ is

1. $\frac{1}{n+1}$
2. $\frac{1}{n+2}$
3. $\frac{1}{n+1}-\frac{1}{n+2}$
4. $\frac{1}{n+1}+\frac{1}{n+2}$

Q. Let F(x) = f(x) + $f\left(\frac{1}{x}\right),f\left(x\right)={\int }_1^x\frac{logt}{1+t}dt.$ Then F(e) equals

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

Q.

${\int }_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{\sqrt{1+cosx}}{(1-cosx{)}^{\frac{5}{2}}}dx=$ [AI CBSE 1980]

1. $\frac{5}{2}$

2. $\frac{3}{2}$

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

4. $\frac{2}{5}$

Q. ${\int }_0^{1}si{n}^{-1}\left(\frac{2x}{1+x^2}\right)dx=$ [Karnataka CET 1999]

1. $\frac{\pi }{2}-2log\sqrt{2}$
2. $\frac{\pi }{2}+2log\sqrt{2}$
3. $\frac{\pi }{4}-log\sqrt{2}$
4. $\frac{\pi }{4}+log\sqrt{2}$

Q. ${\int }_a^b\sqrt{\left[\frac{x-a}{b-x}\right]}dx=$

1. $\frac{2\pi (b-a)}{3}$
2. $\frac{\pi (b+a)}{2}$
3. $\frac{\pi (b-a)}{2}$
4. $\frac{\pi (b-a)}{3}$

Q. ${\int }_0^1\frac{dx}{[ax+b(1-x){]}^2}=$

1. $\frac{a}{b}$
2. $\frac{b}{a}$
3. $ab$
4. $\frac{1}{ab}$

Q. ${\int }_0^{\frac{\pi }{2}}\frac{cosx}{1+sinx}dx=$

1. log 2

2. log e

3. $\frac{1}{2}log3$

4. 0

Q.

${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{e^x.se{c}^{2}xdx}{e^{2x}-1}$is equal to

1. 0

2. 2

3. e

4. 2e

Q. ${\int }_0^{\frac{\pi }{2}}\frac{dx}{2+cosx}=$

1. $\frac{1}{\sqrt{3}}ta{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
2. $\sqrt{3}ta{n}^{-1}\left(\sqrt{3}\right)$
3. $\frac{2}{\sqrt{3}}ta{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
4. $2\sqrt{3}ta{n}^{-1}\left(\sqrt{3}\right)$