List of top Mathematics Questions on Integral Calculus asked in JEE Main

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 \( f\left( \frac{1}{2} \right) \):

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

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:

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.

The number of integral solutions of \( \log_2 \left( \frac{x - 7}{2x - 3} \right) \geq 0 \) is:

If the value of the integral \(\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} dx = \alpha e^{-1} + \beta\), where \(\alpha, \beta \in R, 5\alpha+6\beta=0\), and \([x]\) denotes the greatest integer less than or equal to x; then the value of \((\alpha + \beta)^2\) is equal to :

The value of the integral \( \int_{0}^{1} \frac{\sqrt{x} dx}{(1+x)(1+3x)(3+x)} \) is :

The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1+\sin^2x}{1+\pi^{\sin x}} dx\) is :