List of top Mathematics Questions asked in AP EAPCET

If \( u(n) = \int_0^{\frac{\pi}{2}} (1 + \sin t)^n \sin 2t \, dt,\; n \in \mathbb{N} \), then \( u(4) = \)

If \( f(x) = \int \frac{dx}{x^2 + 2} \) and \( f(\sqrt{2}) = 0 \), then \( f(0) = \)

Assertion (A): \(\displaystyle \int_0^{\frac{\pi}{2}} (\sin^6 x + \cos^6 x)\, dx\) lies in the interval \(\left(\frac{\pi}{8}, \frac{\pi}{2}\right)\) Reason (R): \(\sin^6 x + \cos^6 x\) is a periodic function with period \(\dfrac{\pi}{2}\)

If \( f(x) = \int_0^x \frac{5t^8 + 7t^6}{(t^2 + 2t + 1)^2} dt \) and \( f(0) = 0 \), then the value of \( f(1) \) is:

\( \int_0^{1/2} \frac{x \sin^{-1} x}{\sqrt{1 - x^2}} dx = \)

Evaluate the integral: \[ \int \frac{e^x \left( 2 + \sin(2x) \right)}{1 + \cos(2x)} \, dx \]

Evaluate the integral: \[ \int_0^{\frac{\pi}{4}} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \]

Evaluate the integral: \[ \int \frac{dx}{4 + 5 \cos x} \]

Evaluate the integral: \[ \int_0^{50\pi} \sqrt{1 - \cos 2x} \, dx \]

If $ f(x) = \frac{x}{(1 + nx^n)^{1/n}} $ for $ n \geq 2 $, then \[ \int x^{n-2} f(x) \, dx = \]

If \[ \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} \, dx = Ax + B \log \left( 9e^{2x} - 4 \right) + C, \text{ then } (A, B) = \]

If \( \displaystyle \int_{3}^{b} \frac{x - 1}{2x - x^2} \, dx = \frac{1}{2} \), then \( (b - 1)^2 = \)

The integral \[ \int \frac{1 + x \cos x}{x \left[ 1 - x^2 \left( e^{\sin x} \right)^2 \right]} \, dx \] Options:

If $ a = 2n $ and $ b = 2m+1 $ for all $ m, n \in \mathbb{N} $, then \[ \int_{-\pi}^{\pi} e^{\sin^2 x} \cot^{b}(2n+1) x \, dx = \]

If \( [x] \) is the greatest integer not exceeding \( x \), then \[ \int_{-0.5}^{1.5} x^2 [x] \, dx = \]

If \( a \in \mathbb{Z}^+ \), \([x]\) is the greatest integer not more than \( x \) and \[ \int_0^a \left\lfloor x \right\rfloor \, dx = 127, \text{ then } a = ? \]

The area bounded by the curves \( y - 1 = \cos x \), \( y = \sin x \) and the X-axis between \( x = 0 \) and \( x = \pi \) is

If \( f(x) = \int_0^x \left[ (a+1)(t+1)^2 - (a-1)(t^2 + t + 1) \right] dt \), then a possible positive value of \( a \), for which \( f'(x) = 0 \) has equal roots, is:

Evaluate the integral \[ \int \frac{\sin^{-1} \sqrt{x} - \cos^{-1} \sqrt{x}}{\sqrt{x} \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right)} \, dx \]

If \( \int_0^{2024\pi} \frac{2023^{\sin^2 x}}{2023^{\sin^2 x} + 2023^{\cos^2 x}} dx = k \), then \( \left( \frac{2k}{\pi} + 1 \right) = \)

If \[ \int \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R}, \text{ then} \] \[ \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R} \]

If \( f(x) = \begin{cases} x^2, & 0 \le x<1 \\ \sqrt{x}, & 1 \le x \le 2 \end{cases} \), then \( \int_0^2 f(x) dx = \)

If \(\displaystyle \int \frac{3x+2}{4x^2+4x+5} \, dx = A \log(4x^2+4x+5) + B \tan^{-1} \left(x + \frac{1}{2}\right) + C\), then \((A, B) =\)

\(\displaystyle \int e^x \left( \log x + \frac{1}{x^2} \right) dx =\)