Indefinite Integration — NDA Mathematics PYQs

Directions : For the next two (02) items that follow :
Let $\int \frac{\sin \theta d\theta }{(2+\cos \theta )(3+4\cos \theta )}=A\ln |2+\cos \theta |+B\ln |3+4\cos \theta |$

What is the value of $A$ ?

Directions : For the next two (02) items that follow :
Let $\int \frac{\sin \theta d\theta }{(2+\cos \theta )(3+4\cos \theta )}=A\ln |2+\cos \theta |+B\ln |3+4\cos \theta |$

What is the value of $B$ ?

Consider the following for the two (02) items that follow :
Let the function $y=(1-\cos x{)}^{-1}$, where $x\ne 2n\pi$ and $n$ is an integer.

What is $\int ydx$ equal to?

Directions : For the following two (02) items :
Let $u=\int e^x\cos xdx$ and $v=\int e^x\sin xdx$.

What is $u+v$ equal to?

Directions : For the following two (02) items :
Let $u=\int e^x\cos xdx$ and $v=\int e^x\sin xdx$.

Consider the following :
I. $\frac{du}{dx}=-v$
II. $\frac{dv}{dx}=-u$
Which of the above is/are correct?

What is $\int \frac{dx}{10^x+10^{-x}}$ equal to?

*Directions : Consider the following for the next two (02) items that follow :

Given that $\int \frac{3\cos x+4\sin x}{2\cos x+5\sin x}dx=\frac{\alpha x}{29}+\frac{\beta }{29}\ln |2\cos x+5\sin x|+c$*

What is the value of $\alpha$ ?

*Directions : Consider the following for the next two (02) items that follow :

Given that $\int \frac{3\cos x+4\sin x}{2\cos x+5\sin x}dx=\frac{\alpha x}{29}+\frac{\beta }{29}\ln |2\cos x+5\sin x|+c$*

What is the value of $\beta$ ?

Consider the following for the next two (02) items that follow :

Let $\int \frac{dx}{\sqrt{x+1}-\sqrt{x-1}}=\alpha (x+1{)}^{\frac{3}{2}}+\beta (x-1{)}^{\frac{3}{2}}+c$

What is the value of $\alpha$ ?

Consider the following for the next two (02) items that follow :

Let $\int \frac{dx}{\sqrt{x+1}-\sqrt{x-1}}=\alpha (x+1{)}^{\frac{3}{2}}+\beta (x-1{)}^{\frac{3}{2}}+c$

What is the value of $\beta$ ?