List of top Mathematics Questions on Integral Calculus

Let \(y = y(x)\) be the solution of the differential equation \(x\sqrt{1-x^2} \, dy + (y\sqrt{1-x^2} - x \cos^{-1} x) \, dx = 0\), \(x \in (0, 1)\), \(\lim_{x \to 1^-} y(x) = 1\). Then \(y\left(\frac{1}{2}\right)\) equals:
The value of the integral $\int_0^\infty \frac{\log_e (x)}{x^2 + 4} dx$ is:
If \[ \alpha=\int_{0}^{2\sqrt{3}} \log_2(x^2+4)\,dx + \int_{2}^{4} \sqrt{2^x-4}\,dx, \] then \(\alpha^2\) is equal to _____.
If $\int_{-2}^2 ([\sin x] + |x \sin x|) dx = 2 \sin 2 - 4 \cos 2 - \beta$, then the value of $|\beta|$ where $[\cdot]$ is GIF is
If \( I(x) = \int \frac{16x+24}{x^2+2x-15}\,dx \), \( I(1) = 14\ln 3 \) and \( I(7) = \ln\left(2^\alpha \cdot 3^\beta\right) \), then \( (\alpha+\beta) \) is equal to
The value of \( \int_{0}^{20\pi} \left( \sin^4 x + \cos^4 x \right)\, dx \) is equal to
If the area bounded by two curves \( \dfrac{x^2}{9} - \dfrac{y^2}{16} = 1 \) and \( 8x - 3y = 24 \) is \( A - 6\log_e 3 \), then \( A \) is equal to
Evaluate: \[ \frac{6}{3^{26}}+\frac{10\cdot1}{3^{25}}+\frac{10\cdot2}{3^{24}}+\frac{10\cdot2^{2}}{3^{23}}+\cdots+\frac{10\cdot2^{24}}{3}. \]
If \[ I(x) = 3\int \frac{dx}{(4x+6)\sqrt{4x^2 + 8x + 3}}, \quad I(0) = \frac{\sqrt{3}}{4}, \] then find \( I(1) \):
If \[ \int e^x \left( \frac{x^2 - 2}{\sqrt{1 + x(1 - x)^{3/2}}} \right) \, dx = f(x) + c \quad \text{and} \quad f(0) = 1 \] find \( f\left( \frac{1}{2} \right) \):
The antiderivative of $\dfrac{1}{x\sqrt{x^2-1}},\; x>1$ with respect to $x$ is
The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{7}x \, dx \] is
Given below are two statements: Statement I: \[ 25^{13}+20^{13}+31^{13} \text{ is divisible by } 7 \] Statement II: The integral part of \(\left(7+4\sqrt3\right)^{25}\) is an odd number. In the light of the above statements, choose the correct answer:
If \[ \int_{0}^{x} t^2 \sin(x - t)\,dt = x^2, \] then the sum of values of \( x \), where \( x \in [0,100] \), is:
Find the area bounded by the curves \[ x^2 + y^2 = 4 \quad \text{and} \quad x^2 + (y-2)^2 = 4. \]
The value of \[ \int_{\frac{\pi}{2}}^{\pi} \frac{dx}{[x]+4} \] where \([\,\cdot\,]\) denotes the greatest integer function, is
In a triangle with one of the angles \( 120^\circ \), the lengths of the sides form an A.P. If the length of the greatest side is 7 m, then the area of the triangle is
The area bounded by the curve \( y = x^2 + 3, y = x, x = 3 \) and \( y \)-axis is
If \( \theta \) is the angle between the lines whose direction cosines are given by \( 6mn - 2nl + 5lm = 0 \) and \( 3l + m + 5n = 0 \), then \( \sin \theta = \)}
If \( A = \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \end{bmatrix} \), \( B = \text{adj } A \) and \( C = 5A \), then \( \frac{|\text{adj } B|}{|C|} = \)}
The Cartesian equation of plane through A$(7, 8, 6)$ and parallel to the XY plane is
The RMS value of \(x^2\) in \([0,1]\) is
The area bounded by the curve \(y=4x^2\), the \(x\)-axis, the line \(x=0\) and the line \(x=1\) is
\[ \int_0^{\pi/2}\frac{\cos2x}{\sin x+\cos x}\,dx= \]
The value of \[ \int_0^1 x(1-x)^9\,dx \] is