List of top Mathematics Questions on Integral Calculus

Evaluate: \[ \frac{6}{3^{26}}+\frac{10\cdot1}{3^{25}}+\frac{10\cdot2}{3^{24}}+\frac{10\cdot2^{2}}{3^{23}}+\cdots+\frac{10\cdot2^{24}}{3}. \]
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 \[ \int e^x \left( \frac{x^2 - 2}{\sqrt{1 + x(1 - x)^{3/2}}} \right) \, dx = f(x) + c \quad \text{and} \quad f(0) = 1 \] find \( f\left( \frac{1}{2} \right) \):
If \[ I(x) = 3\int \frac{dx}{(4x+6)\sqrt{4x^2 + 8x + 3}}, \quad I(0) = \frac{\sqrt{3}}{4}, \] then find \( I(1) \):
Find the area bounded by the curves \[ x^2 + y^2 = 4 \quad \text{and} \quad x^2 + (y-2)^2 = 4. \]
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:
The value of \[ \int_{\frac{\pi}{2}}^{\pi} \frac{dx}{[x]+4} \] where \([\,\cdot\,]\) denotes the greatest integer function, 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) $.)
\(\tan^{-1}(1) - \sec^{-1}2)\) is equal to:(
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:
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 \( f(x) \)?
The value of the integral \( \int_{3}^{6} \frac{\sqrt{x}}{\sqrt{9 - x} + \sqrt{x}} \, dx \) is:
If \( x = \int_0^y \frac{1}{\sqrt{1+9t^2}} \, dt \) and \( \frac{d^2y}{dx^2} = ay \), then the value of \( a \) is:
Evaluate the integral \( \int_{-1}^{1} \frac{x^2 + |x| + 1}{x^2 + 2|x| + 1} \, dx \):
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) $.)
Maximum value of the function \( f(r) = r^2 e^{-r} \), when \( 0<r<\infty \) is
The value of the derivative of the function \( y = x e^x \) at \( x = 1 \) is ........... (rounded off to two decimal places).
Let the function \(f:[1,\infin)→\R\) be defined by
\(f(t) = \begin{cases}     (-1)^{n+1}2, & \text{if } t=2n-1,n\in\N, \\     \frac{(2n+1-t)}{2}f(2n-1)+\frac{(t-(2n-1))}{2}f(2n+1) & \text{if } 2n-1<t<2n+1,n\in\N. \end{cases}\)
Define \(g(x)=\int\limits_{1}^{x}f(t)dt,x\in(1,\infin).\) Let α denote the number of solutions of the equation g(x) = 0 in the interval (1, 8] and \(β=\lim\limits_{x→1+}\frac{g(x)}{x-1}\). Then the value of α + β is equal to _____.
Let $\beta(m, n) = \int_{0}^{1}x^{m-1}(1-x)^{n-1}dx$, $m, n > 0$. If $\int_{0}^{1}(1-x^{10})^{20}dx = a\beta(b,c)$, then $100(a+b+x)$ equals
If \[ \int \frac{1}{\sqrt[5]{(x - 1)^4}(x + 3)^6} \, dx = A \left( \frac{\alpha x - 1}{\beta x + 3} \right)^B + C, \] where \(C\) is the constant of integration, then the value of \(\alpha + \beta + 20AB\) is _______.
Let \( \int \frac{2 - \tan x}{3 + \tan x} \, dx = \frac{1}{2} \left( \alpha x + \log_e \lvert \beta \sin x + \gamma \cos x \rvert \right) + C \), where \( C \) is the constant of integration.