Differentiation under Integral Sign

Q. ${\int }_{-2\pi }^{2\pi }\frac{si{n}^{6}x}{(si{n}^{6}x+co{s}^{6}x)(1+e^{-x})}dx=$

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

Q. If $f$ is strictly increasing and positive function, such that $x\underset{0}{\overset{x}{\int }}(1-t)\sin \left(f\right(t\left)\right)dt=2\underset{0}{\overset{x}{\int }}t\sin \left(f\right(t\left)\right)dt,$ where $x>0.$ Then the value of $f^{\prime }\left(x\right)\cot f\left(x\right)+\frac{3}{1+x}$ in the domain of $f\left(x\right)$ is

Q. If $f$ is continuous function and $F\left(x\right)={\int }_0^x\left(\right(2t+3\left){\int }_t^{2}f\left(u\right)du\right)dt$, then the value of $\left|\frac{F^{\prime \prime }\left(2\right)}{f\left(2\right)}\right|$

1. 9
2. 3
3. 5
4. 7

Q. If $f\left(x\right)$ is differentiable and ${\int }_0^{t^2}xf\left(x\right)dx=\frac{2}{5}{t}^5,\text{then}f\left(\frac{4}{25}\right)$ equals

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

Q.

The value of $\underset{x\to 0}{lim}\frac{1}{x^3}{\int }_0^x\frac{t\ln (1+t)}{t^4+4}dt$ is

1. $0$

2. $\frac{1}{12}$
   

3. $\frac{1}{24}$

4. $\frac{1}{6}$

Q. $\underset{x\to \frac{\pi }{4}}{lim}\frac{{\int }_2^{se{c}^{2}x}f\left(t\right)dt}{x^2-\frac{{\pi }^2}{16}}$equals

1. $\frac{8}{\pi }f\left(2\right)$
2. $\frac{2}{\pi }f\left(\frac{1}{2}\right)$
3. $4f\left(2\right)$
4. $\frac{2}{\pi }f\left(2\right)$

Q. Let $f:R\to R$ be a differentiable function having $f\left(2\right)=6,f^{\prime }\left(2\right)=\left(\frac{1}{48}\right)$. Then $\underset{x\to 2}{lim}{\int }_6^{f\left(x\right)}\frac{4t^3}{x-2}dt$ equals

1. 18
2. 24
3. 12
4. 36

Q.

The value of ${\int }_0^{si{n}^{2}x}si{n}^{-1}\left(\sqrt{t}\right)dt+{\int }_0^{co{s}^{2}x}co{s}^{-1}\left(\sqrt{t}\right)dt$ is

1. 0

2. None of these

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

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

Q. If $x={\int }_0^y\frac{\mathrm{dt}}{\sqrt{1+9t^2}}\mathrm{and}\frac{d^{2}y}{{\mathrm{dx}}^2}=\mathrm{ay}$, then the value of a is equal to

Q. Let $f\left(x\right)={\int }_1^x\sqrt{2-t^2}\mathrm{dt}$. Then the real roots of the equation $x^2-f\left(x\right)=0$ are

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