Definite Integration — NDA Mathematics PYQs

Directions : For the next two (02) items that follow :
Let ${\int }_0^{\frac{\pi }{2}}\frac{a\sin x+b\cos x}{(a+b)(\sin x+\cos x)}dx=k$

What is the value of $k$ ?

Directions : For the next two (02) items that follow :
Let ${\int }_0^{\frac{\pi }{2}}\frac{a\sin x+b\cos x}{(a+b)(\sin x+\cos x)}dx=k$

What is ${\int }_0^{\frac{\pi }{2}}\frac{a\cos x+b\sin x}{\sin x+\cos x}dx$ equal to ?

Directions : For the next two (02) items that follow :
Let $f(x)=\sin x$ and $g(x)-f(x)=f(4-x)$.

What is ${\int }_0^4\frac{f(x)}{g(x)}dx$ equal to ?

Directions : For the next two (02) items that follow :
Let $f(x)=\sin x$ and $g(x)-f(x)=f(4-x)$.

What is ${\int }_0^4\frac{f(4-x)}{g(4-x)}dx$ equal to ?

Consider the following for the two (02) items that follow :
Let $f(x)=[x^2]$, where $[\cdot ]$ is the greatest integer function.

What is ${\int }_{\sqrt{2}}^{\sqrt{3}}f(x)dx$ equal to?

Consider the following for the two (02) items that follow :
Let $f(x)=[x^2]$, where $[\cdot ]$ is the greatest integer function.

What is ${\int }_{\sqrt{2}}^{2}f(x)dx$ equal to?

What is ${\int }_n^{n+1}(x-[x])dx$, where $[\cdot ]$ is the greatest integer function and $n$ is natural number?

If $I_1={\int }_e^{e^2}\frac{dx}{\ln x}$ and $I_2={\int }_1^2\frac{e^x}{x}dx$, then which one of the following is correct?

What is ${\int }_{-1}^1(3\sin x-\sin 3x){\cos }^{2}xdx$ equal to ?

Let $p={\int }_a^{b}f(x)dx$ and $q={\int }_a^b|f(x)|dx$. If $f(x)=e^{-x}$, then which one of the following is correct ?