Property 5

Q. Let ${\int }_0^x\left(\frac{btcos4t-asin4t}{t^2}\right)dt=\frac{asin4x}{x}$, then a and b can be

1. $\frac{1}{4},1$
2. 2, 2
3. -1, 4
4. 2, 4

Q. If $f\left(x\right)={\int }_1^x\frac{dt}{2+t^4}$, then

1. $f\left(2\right)<\frac{1}{3}$
2. $f\left(2\right)>\frac{1}{3}$
3. $f\left(2\right)\frac{1}{3}$
4. $f\left(2\right)>1$

Q. If $f\left(x\right)={\int }_0^x(1+t^3{)}^{\frac{-1}{2}}$ and g (x) is the inverse of f, then the value of $\frac{g"\left(x\right)}{g^2\left(x\right)}$ is

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

Q. If f(x) is a differentiable function and ${\int }_0^{x^3}{t}^{2}f\left(t\right)dt=\frac{3}{13}{x}^{13}+5$ then $f\left(\frac{8}{27}\right)$

1. $\frac{8}{27}$
2. $\frac{16}{27}$
3. $\frac{16}{81}$
4. $\frac{8}{9}$

Q. The value of the integral $\underset{0}{\overset{\pi }{\int }}|\sin 2x|dx$ is

Q. If f(x) can be written as $f\left(x\right)={\int }_1^{x-1}{e}^{y^2}dy$ then which of the following is true about f(x)?

1. f(x) is monotonically increasing in the interval [0, 1]
2. f(x) is strictly increasing in the interval [0, 1]
3. f(x) is monotonically increasing for all real values of x.
4. f(x) is strictly increasing for all real values of x.

Q. $If{\int }_0^{x}f\left(t\right)dt=x+{\int }_x^{1}tf\left(t\right)dt,$ then the value of f (1) is

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

Q. The value of $li{m}_{x\to 0}\frac{{\int }_0^{x^2}se{c}^{2}tdt}{xsinx}$ is

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

Q. ${\int }_0^{2a}f\left(x\right)dx={\int }_0^a\left(f(a-x)+f(a+x)\right\}dx$

1. True
2. False

Q. Let $f:(0,\infty )$ and F(x)=${\int }_1^{x}f\left(t\right).IfF\left(x^2\right)=x^2(1+x)$ then f(4) equals

1. $\frac{5}{4}$
2. $7$
3. $4$
4. $2$

Q. For $xϵ(0,\frac{5\pi }{2}),definef\left(x\right)={\int }_0^x\sqrt{t}sintdt.$ Then f has

1. local minimum at $\pi$ and 2 $\pi$
2. local minimum at $\pi$ and local maximum at 2$\pi$
3. local maximum at $\pi$ and local minimum at 2$\pi$
4. local maximum at $\pi$ and 2$\pi$

Q.

If $f\left(x\right)={\int }_a^x{t}^3{e}^{t}dt,then\frac{d}{dx}f\left(x\right)=$ [MP PET 1989]

1. $e^x(x^3+3x^2)$
2. $x^3{e}^x$

3. $a^3{e}^x$

4. $Noneofthese$

Q. Which of the following pair(s) of functions is/are not identical

1. $f\left(x\right)=\cos x,g\left(x\right)=\frac{1}{\sec x}$
2. $f\left(x\right)=\frac{|x|}{x},g\left(x\right)=1$
3. $f\left(x\right)=\sin x,g\left(x\right)=\frac{1}{\text{cosec}x}$
4. $f\left(x\right)=\text{sgn}\left(x\right),g\left(x\right)=\frac{|x|}{x}$

Q. If $f\left(x\right)={\int }_0^x(1+t^3{)}^{\frac{-1}{2}}$ and g (x) is the inverse of f, then the value of $\frac{g"\left(x\right)}{g^2\left(x\right)}$ is

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