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 value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{7}x \, dx \] is
The antiderivative of $\dfrac{1}{x\sqrt{x^2-1}},\; x>1$ with respect to $x$ 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
The value of the integral \( \int_{3}^{6} \frac{\sqrt{x}}{\sqrt{9 - x} + \sqrt{x}} \, dx \) is:
If $$ \alpha = \int_{\frac{1}{2}}^{2} \frac{\tan^{-1} x}{2x^2 - 3x + 2} \, dx, $$ then the value of $ \sqrt{7} \tan \left( \frac{2\alpha \sqrt{7}}{\pi} \right) $ is.
(Here, the inverse trigonometric function $ \tan^{-1} x $ assumes values in $ \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) $.)
If the radius of a circle increases at a uniform rate of \(2\ \text{cm/s}\), then the rate of increase of area of the circle, at the approximate instant when the radius is \(20\ \text{cm}\), is:
\(\tan^{-1}(1) - \sec^{-1}2)\) is equal to:(
If \(\cos x = \tan y,\ \cot y = \tan z\) and \(\cot z = \tan x\), then \(\sin x\) is equal to:
Find the value of the integral: \[ \int_0^e \log_e x \, dx \]
If \[ 24 \left( \int_0^\frac{\pi}{4} \left[ \sin \left( 4x - \frac{\pi}{12} \right) + [2 \sin x] \right] dx \right) = 2n + \alpha, \] where [.] denotes the greatest integer function, then \( \alpha \) is equal to:
Find the value of the integral \( \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx \).
For the function \( f(x) = \ln(x^2 + 1) \), what is the second derivative of \( f(x) \)?