List of top Mathematics Questions on Some Properties of Definite Integrals

$\int_{0}^{\pi/2} \log\left( \frac{\cos x}{\sin x} \right)\, dx$ is equal to:
If \( f(a + b - x) = f(x), \) then \( \int_a^b f(x) \, dx \) is
What is the area of a loop of the curve \( y = \sin 30^\circ \)?
\( \int_{-\pi}^{\pi} \frac{\sin^2 x}{1+7^x}\, dx \) is equal to
\( \int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} \, dx \) is equal to
If \( (m,n) \) is an integer solution of \( \int_1^\infty \left( 1 + y^2 \right) dy \), then the expression for \( (m, n) \) in terms of \( (m+1, n+1) \) is?
\( \int_0^\infty \left( \sin x + \left| \sin x \right| \right) dx \) is equal to?
The value of \( \int_0^\infty \left[ \sqrt{x} \right] dx \), where \( \left[ . \right] \) is the greatest integer function, is?
The value of the integral $\int_\limits{-\pi/4}^{\pi/4} \log (\sec \theta-\tan \theta) d \theta$ is
The pair of lines \(\sqrt{3}x^2 - 4xy + \sqrt{3}y^2 = 0\) are rotated about the origin by \(\frac{\pi}{6}\) in the anticlockwise sense. The equation of the pair in the new position is
The value of the integral $\int\limits ^{1/2}_{-1/2}\left[\left(\frac{x+1}{x-1}\right)^{^2}+\left(\frac{x+1}{x-1}\right)^{^2}-2\right]^{^{1/2}}\:\:dx$ is
The area lying in the first quadrant and bounded by the circle \( x^2 + y^2 = 4 \), the line \( x = \sqrt{3}y \), and the \( x \)-axis is:
Let \[ I_1 = \int_{\alpha}^{\pi - \alpha} x \sin x \, dx, I_2 = \int_{\alpha}^{\pi - \alpha} f(\sin x) \, dx \] then \( I_2 \) is equal to:
$\int\limits^2_0 |1 -x| dx$ is equal to
The value of the integral $ \int{\frac{1}{{{e}^{2x}}+{{e}^{-2x}}}}\,\,dx $ is equal to
The value(s ) of $ \int^1_0 \frac { x^4 ( 1 - x )^4 }{ ( 1 + x^2 ) } \, dx $ is are
The value of \( \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx \) is:
The value of the integral $\int\limits_0^{\pi/2}$($Sin^{100} x-Cos^{100}x)dx$ is
The value of the integral $\int\limits_{0}^{\frac{\pi}{2}} log \left(tan\,x\right)dx=$
The value of the integral $ \int^{ \pi / 2 }_ 0 \frac {\sqrt { \cot \, x }}{ \sqrt { \cot \, x } + \sqrt { \tan \, x }} \, dx \, is $
Evaluate: \[ \int_1^3 (|x - 1| + |x - 2| + |x - 3|) \, dx. \]
\( \int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} \, dx \) is equal to:
Evaluate: \[ \int_{-2}^{2} \frac{x^3 + |x| + 1}{x^2 + 4|x| + 4} \, dx. \]
Evaluate: \[ \int_0^{\frac{\pi^2}{4}} \frac{\sin\sqrt{x}}{\sqrt{x}} \, dx \]
If \( f(x) \) is an odd function, then: \[ \int_{-\pi/2}^{\pi/2} f(x) \cos^3 x \, dx \, \text{equals:} \]