Property 1

Q. For $n>0,{\int }_0^{2\pi }\frac{xsi{n}^{2n}x}{si{n}^{2n}x+co{s}^{2n}x}dx=$

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

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. ${\int }_0^1\frac{ta{n}^{-1}x}{1+x^2}dx=$

1. $\frac{{\pi }^2}{8}$
2. $\frac{{\pi }^2}{16}$
3. $\frac{{\pi }^2}{4}$
4. $\frac{{\pi }^2}{32}$

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. ${\int }_0^{\frac{\pi }{2}}\frac{cosx}{(1+sinx)(2+sinx)}dx=$ [UPSEAT 1999]

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

Q. ${\int }_0^{\frac{\pi }{2}}\frac{3secx+5cosecx}{secx+cosecx}dx=$

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

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. ${\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}cos\theta cose{c}^2\theta d\theta =$

1. $\sqrt{2}-1$
2. $1-\sqrt{2}$
3. $\sqrt{2}+1$
4. $Noneofthese$

Q. ${\int }_0^{\frac{1}{\sqrt{2}}}\frac{si{n}^{-1}x}{(1-x^2{)}^{\frac{3}{2}}}dx=$

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

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. Evaluate ${\int }_0^{\pi /2}si{n}^{-1}\left(cosx\right)dx$

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

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

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

Q.

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

1. 0

2. 2

3. e

4. 2e

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

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. ${\int }_{\frac{-3\pi }{2}}^{-\frac{\pi }{2}}\left[\right(x+\pi {)}^3+co{s}^2(x+\pi )]dx$ is equal to

1. $\left(\frac{{\pi }^4}{32}\right)+\left(\frac{\pi }{2}\right)$
2. $\frac{\pi }{2}$
3. $-1$
4. $\frac{{\pi }^4}{32}$

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

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

Q.

The value of ${\int }_{-1}^1(2\left|x\right|-{\left|x\right|}^3)dx$ is

1. 0

2. 3/4

3. 3/2

4. 3

Q. ${\int }_0^{\frac{1}{2}}\frac{xsi{n}^{-1}x}{\sqrt{1-x^2}}dx=$

1. $\frac{1}{2}+\frac{\sqrt{3}\pi }{12}$
2. $\frac{1}{2}-\frac{\sqrt{3}\pi }{12}$
3. $\frac{1}{2}-\frac{\sqrt{3}\pi }{12}$
4. $Noneofthese$

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. ${\int }_0^{\pi }\frac{tanx}{secx+cosx}dx=$

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