List of top Questions asked in AP EAPCET

\( \lim_{n \rightarrow \infty} \frac{1}{n}\sum_{r=1}^{n}\sin^{k}\left(\frac{\pi r}{2n}\right)\cos\left(\frac{\pi r}{2n}\right) = \)

Let \[ f(x)=\int \frac{\sqrt{x}}{(1+x)^2}\,dx \quad (x\geq 0) \] Then \[ f(3)-f(1) \] is equal to:

Evaluate: \[ \int_{\pi/2}^{4051\pi/2}\frac{\cos^22x}{1+\sin2x}\,dx \]

The value of the integral \( \int_{0}^{4} ||x - 2| - x| dx = \)

If \( [ ] \) denotes the greatest integer function, then \( \int_{1}^{2} [x^2] dx = \)

The value of the integral \(\int_{0}^{\pi} \frac{x \sin x}{\sin^2 x + 2 \cos^2 x} \, dx\) is:

Evaluate: \[ \int_{-2\pi}^{2\pi}(1+\cos x)^3(1-\cos x)^4\,dx \]

The value of the definite integral $\int_0^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} \, dx$ is:

The integral \( \int_{0}^{\pi} |x \cos 2x| dx \) is equal to:

The value of the definite integral $\int_{-\pi/2}^{\pi/2} (x^3 + x\cos x + \tan^5 x + 1) \, dx$ is:

The value of the integral \(\int_{0}^{2} x^2 (2 - x)^5 \, dx\) is:

The value of the integral \[ \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} \frac{\sin^4(4x)}{1 + e^{4x}} \, dx \] is equal to:

The value of the integral \[ \int_{0}^{2} x^8 \left( \frac{4}{x^2} - 1 \right)^{\frac{5}{2}} \, dx \] is equal to:

The value of the integral \(\int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx\) is:

Let the function $ f(x) $ be defined as $ f(x) = \frac{x - |x|}{x} $, then:

If \[ \int \frac{dx}{x^4+1} = \frac{1}{4\sqrt{2}} \log \left( \frac{x^2+\sqrt{2}x+1}{x^2-\sqrt{2}x+1} \right) + \frac{1}{2\sqrt{2}} \tan^{-1} \left( \frac{\sqrt{2}x}{1-x^2} \right) +c, \] then \[ \int_{0}^{1}\frac{x^2+1}{x^4+1}\,dx = \]

The integral \( \int_{\sqrt{2}}^{2} \frac{x}{(x^3 - x^2 + x - 1)(x+1)} dx \) is equal to:

The value of the integral \[ \int_{-2}^{2} x^4 (4 - x^2)^{\frac{7}{2}} \, dx \] is equal to:

The value of the integral \(\int_{5}^{9} \frac{\log(3x^2)}{\log(3x^2) + \log(588 - 84x + 3x^2)} \, dx\) is equal to:

If \( \int_{\pi/4}^{3\pi/4} \frac{x \sin x}{1+3\cos 2x} \, dx = k \int_{\pi/4}^{3\pi/4} \frac{\sin x}{1+3\cos 2x} \, dx \), then \( \int_{0}^{k} \sin^{\pi/k}x \, dx = \)

The integral \( \int_{0}^{1} x^{5/2} (1-x)^{3/2} dx \) is equal to:

If $f(x)$ is a twice differentiable function and $f^{\prime}(0)=0$, then $\int_{0}^{\pi/2}(f(x)+f^{\prime\prime}(x))\cos x dx=$

The value of the definite integral $\int_0^{\pi/2} \ln(\tan x) \, dx$ is:

The value of the integral \(\int_{0}^{3\pi/2} \frac{\cos^3 x}{\cos^3 x + \sin^3 x} \, dx\) is:

If \( A = \begin{bmatrix} a & 1 & 2 1 & 2 & b c & 1 & 3 \end{bmatrix} \) and \( \text{adj } A = \begin{bmatrix} 7 & -1 & -5 -3 & 9 & 5 1 & -3 & 5 \end{bmatrix} \), then \( a^2 + b^2 + c^2 = \)