List of top Mathematics Questions on Integration asked in AP EAMCET

The parametric form of a curve is \(x=\dfrac{t^3}{t^2-1},\; y=\dfrac{t}{t^2-1}\), then \(\displaystyle \int \frac{dx}{x-3y}=\)

Assertion (A): If \(I_n=\int \cot^n x \, dx\), then \(I_6+I_4=\dfrac{-\cot^5 x}{5}\)
Reason (R): \[ \int \cot^n x \, dx=\frac{-\cot^{\,n-1}x}{n-1}-\int \cot^{\,n-2}x \, dx \]

If
\[ I_n=\int \tan^n x\,dx \] and
\[ I_0+I_1+2I_2+2I_3+2I_4+I_5+I_6=\sum_{k=1}^{n}\frac{\tan^k x}{k}, \] then \(n=\)

Evaluate
\[ \int \frac{e^{\cot x}}{\sin^2 x}\left(2\log\cosec x+\sin2x\right)\,dx \]