Integral as Antiderivative

Q. If $f\left(x\right)=\underset{0}{\overset{x}{\int }}{e}^t\sin (x-t)dt,$ then $f^{\prime \prime }x-f\left(x\right)$ is equal to

1. $\sin x-\cos x$
2. $2\sin x$
3. $\sin x+\cos x$
4. $2\cos x$

Q. Let $f\left(x\right)$ be a differentiable function satisfying $f\left(\frac{x+y}{2}\right)=\frac{f\left(x\right)+f\left(y\right)}{2}\forall x,y\in R$, where $f\left(0\right)=0$ and $I=\underset{0}{\overset{2\pi }{\int }}{\left(f\right(x)-\sin x)}^{2}dx.$
Which of the following is/are correct?

1. When $I$ is minimum, then $f(–4{\pi }^2)=3$
2. $f\left(x\right)$ is a odd function
3. $I$ is constant.
4. When $I$ is minimum, then $f(–4{\pi }^2)=-3$

Q. If $f\left(x\right)$ be a function such that $\left(f\right(x){)}^{2007}=\underset{0}{\overset{x}{\int }}\frac{\left(f\right(t){)}^{2006}}{2+t^2}dt,$ then which of the following is/are true?

1. There are only two such functions are possible.
2. There are infinite number of such functions are possible.
3. $f\left(x\right)=\frac{1}{2007}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)$
4. $f\left(x\right)=\frac{1}{2007\sqrt{2}}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)$

Q. If $f\left(x\right)$ be a function such that $\left(f\right(x){)}^{2007}=\underset{0}{\overset{x}{\int }}\frac{\left(f\right(t){)}^{2006}}{2+t^2}dt,$ then which of the following is/are true?

1. There are only two such functions are possible.
2. There are infinite number of such functions are possible.
3. $f\left(x\right)=\frac{1}{2007}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)$
4. $f\left(x\right)=\frac{1}{2007\sqrt{2}}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)$

Q. If $f\left(x\right)=\underset{0}{\overset{x}{\int }}{e}^t\sin (x-t)dt,$ then $f^{\prime \prime }x-f\left(x\right)$ is equal to

1. $\sin x-\cos x$
2. $2\sin x$
3. $\sin x+\cos x$
4. $2\cos x$

Q. Let $f\left(x\right)$ be a differentiable function satisfying $f\left(\frac{x+y}{2}\right)=\frac{f\left(x\right)+f\left(y\right)}{2}\forall x,y\in R$, where $f\left(0\right)=0$ and $I=\underset{0}{\overset{2\pi }{\int }}{\left(f\right(x)-\sin x)}^{2}dx.$
Which of the following is/are correct?

1. When $I$ is minimum, then $f(–4{\pi }^2)=3$
2. $f\left(x\right)$ is a odd function
3. $I$ is constant.
4. When $I$ is minimum, then $f(–4{\pi }^2)=-3$