List of top Mathematics Questions on Integration

If $I_1 = \int \frac{e^x}{e^{4x}+e^{2x}+1}dx$, $I_2 = \int \frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx$, then $I_2-I_1=$
Consider the following
Assertion (A): $\int \sqrt{x-3}(\sin^{-1}(\log x) + \cos^{-1}(\log x))dx = \frac{\pi}{3}(x-3)^{3/2}+c$
Reason (R): $\sin^{-1}(f(x))+\cos^{-1}(f(x))=\frac{\pi}{2}$, $|f(x)|\le 1$
The correct answer is
If $\int \frac{3x+2}{4x^2+4x+5}dx = A\log(4x^2+4x+5)+B\tan^{-1}(\frac{2x+1}{2})+c$, then $A+B=$
If $\int \frac{\sqrt{1-\sqrt{x}}}{\sqrt{x(1+\sqrt{x})}}dx = 2f(x)-2\sin^{-1}\sqrt{x}+c$, then $f(x)=$
$\lim_{n \to \infty} \frac{(2n(2n-1)...(n+2)(n+1))^{1/n}}{n} =$
$\int_{-2}^{4} |2-x^2| dx =$
$\int_{0}^{\pi/4} \frac{\sec x}{3\cos x + 4\sin x} dx =$
$\int \frac{3^x(x\log 3 - 1)}{x^2} dx =$
$\int_{-1}^{1} \frac{\log 2 - \log(1+x)}{\sqrt{1-x^2}} dx =$
If $\frac{5\pi}{4}<x<\frac{7\pi}{4}$, then $\int \sqrt{\frac{1-\sin 2x}{1+\sin 2x}} dx =$
If $[\cdot]$ denotes the greatest integer function, then $\int_1^2 [x^2] dx =$
$\int_4^{18} \frac{1}{(x+2)\sqrt{x-3}}dx = $
$\int_0^{\pi/2} \frac{1}{5\cos^2 x + 16\sin^2 x + 8\sin x \cos x} dx =$
$\int \frac{\log x}{(1+x)^2}dx = $
$\int \frac{x\text{Tan}^{-1}x}{(1+x^2)^2}dx =$
$\int(1+\tan^2 x)(1+2x\tan x)dx =$
$\int e^x(x^3-2x^2+3x-4)dx =$
Let \( A = [a_{ij}] \) be a square matrix of order 2 with entries either 0 or 1. Let \( E \) be the event that \( A \) is an invertible matrix. Then the probability \( P(E) \) is:
The integral $ \int_0^\pi \frac{(x + 3) \sin x}{1 + 3 \cos^2 x} \, dx $ is equal to:
The value of \[ \int_{e^2}^{e^4} \frac{1}{x} \left( \frac{e^{\left( (\log_e x)^2 +1 \right)^{-1}}}{e^{\left( (\log_e x)^2 +1 \right)^{-1}} + e^{\left( (6-\log_e x)^2 +1 \right)^{-1}}} \right) dx \] is:

Let (a, b) be the point of intersection of the curve \(x^2 = 2y\) and the straight line \(y - 2x - 6 = 0\) in the second quadrant. Then the integral \(I = \int_{a}^{b} \frac{9x^2}{1+5^{x}} \, dx\) is equal to:

If the function \[ f(x) = \begin{cases} \frac{2}{x} \left( \sin(k_1 + 1)x + \sin(k_2 -1)x \right), & x<0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e \left( \frac{2 + k_1 x}{2 + k_2 x} \right), & x>0 \end{cases} \] is continuous at \( x = 0 \), then \( k_1^2 + k_2^2 \) is equal to:

\(\int \frac{\sqrt{x - 2}}{2x + 4} \, dx =\)
If \( \int x^2 \cos^3 x\, dx = \frac{1}{6}f(x) + g(x) \sin 2x + h(x) \cos 2x + c \), then \( f(1) + g(2) + h\left(\frac{1}{2}\right) = \)
Evaluate: $$ \int_0^1 x \sin^{-1} x \, dx $$