List of top Mathematics Questions on Integration asked in AP EAMCET

Evaluate the integral: \[ I = \int_{-\pi}^{\pi} \frac{x \sin x}{1 + \cos^2 x} \,dx. \]
If \[ \int \frac{\sqrt[4]{x}}{\sqrt{x} + \sqrt[4]{x}} \, dx = \frac{2}{3} \left[ A \sqrt[4]{x^3} + B \sqrt[4]{x^2} + C \sqrt[4]{x} + D \log \left( 1 + \sqrt[4]{x} \right) \right] + K \] then \( \frac{2}{3} (A + B + C + D) = \)}
Evaluate \( \int_{1}^{2} \frac{x^4 - 1}{x^6 - 1} dx \):
Evaluate the limit: $$ \lim_{n \to \infty} n^4 \left[ \sum_{k=0}^{\infty} \frac{1}{(n^2 + k)^{5/2}} \right]. $$
If \(I_{n}=\int_{0}^{\pi/4}\tan^{n}x~dx,\) then \(I_{13}+I_{11}=\)
Evaluate the integral: $$ \int \frac{x^3 \tan^{-1}(x^4)}{1 + x^8} \,dx. $$ 
The area (in square units) of the smaller region lying above the X-axis and bounded between the circle \[ x^2 + y^2 = 2ax \] and the parabola \[ y^2 = ax \]
Evaluate \( \int (\log x)^m x^n dx \).
\(\int\frac{2x^{2}\cos(x^{2})-\sin(x^{2})}{x^{2}}dx=\)

Evaluate the integral: $ I = \int e^{2x+3} \sin 6x \, dx $.

\(\int_{0}^{\pi}x \sin^4 x \cos^6 x dx = \)
Evaluate the integral: \[ I = \int \frac{x+1}{\sqrt{x^2 + x + 1}} dx. \]

Evaluate the integral: $$ I = \int \frac{2}{1 + x + x^2} \,dx. $$ 

Evaluate the integral: \[ I = \int \frac{\sin 7x}{\sin 2x \sin 5x} \,dx. \]
Evaluate the integral: \[ I = \int \frac{1}{x^2\sqrt{1 + x^2}} \,dx. \]
Evaluate the integral: \[ \int \frac{1}{x^5 \sqrt{x^5+1}} dx. \]
Evaluate the integral: \[ \int \sin^4 x \cos^4 x \, dx. \]
If \(\int \frac{\log(1+x^4)}{x^3} dx = f(x) \log(\frac{1}{g(x)}) + \tan^{-1}(h(x)) + c\), then \(h(x) [f(x) + f(\frac{1}{x})] =\)
The set of all real values of \( a \) such that the function \( f(x) = x^3 + 2ax^2 + 3(a+1)x + 5 \) is strictly increasing in its entire domain is:
Evaluate the integral: \[ \int \frac{x^2 - 1}{x^3\sqrt{2x^4 - 2x^2 + 1}} \,dx. \]
Evaluate the integral: \[ I = \int_0^{\frac{\pi}{4}} \log(1 + \tan x) \,dx. \]
Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{\sqrt{n^2}} + \frac{1}{\sqrt{n^2 - 1}} + \dots + \frac{1}{\sqrt{n^2 - (n-1)^2}} \right). \]
Evaluate the integral \( \int_0^{\pi} \frac{x \sin x}{4 \cos^2 x + 3 \sin^2 x} dx \):
Evaluate the integral: \[ \int_{-2}^{2} (4 - x^2)^{\frac{5}{2}} \, dx. \]
Evaluate the integral: \[ \int e^{4x^2 + 8x -4} (x+1) \cos(3x^2 + 6x -4) \, dx.= \]