List of top Mathematics Questions on Definite Integral

If $\sqrt{3}i, 1$ are the roots of $x^{3} + ax^{2} + bx + c = 0$ where $a, b, c \in \mathbb{R}$. Then $\int_{-1}^{1} (x^{3} + ax^{2} + bx + c)dx$ is

The value of $ \int_0^{2\pi} (\sin^4 x + \cos^4 x) \, dx $ is equal to:

Evaluate the definite integral: $ \int_{-2}^{2} |x^2 - x - 2| \, dx $

If the value of the integral

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{x^2 \cos x}{1 + \pi^x} + \frac{1 + \sin^2 x}{1 + e^{\sin^x 2023}} \right) dx = \frac{\pi}{4} (\pi + a) - 2, \]

then the value of $a$ is:

Evaluate the limit : $\lim_{x \to \frac{\pi}{2}} \left( \frac{1}{\left( x - \frac{\pi}{2} \right)^3} \int_{\frac{\pi}{2}}^x \cos \left( \frac{1}{t^3} \right) \, dt \right)$

The value $ 9 \int_{0}^{9} \left\lfloor \frac{10x}{x+1} \right\rfloor \, dx $, where $ \left\lfloor t \right\rfloor $ denotes the greatest integer less than or equal to $ t $, is ________.

For $ x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) $, if\[y(x) = \int \frac{\csc x + \sin x}{\csc x \sec x + \tan x \sin^2 x} \, dx\]and\[\lim_{x \to -\frac{\pi}{2}} y(x) = 0\]then $ y\left(\frac{\pi}{4}\right) $ is equal to

The value of the integral \[ \int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) \, dx \] is:

The value of $\lim_{{n \to \infty}} \sum_{{k=1}}^{n} \frac{n^3}{{(n^2 + k^2)(n^2 + 3k^2)}}$ is

Evaluate $ \int_0^{\pi/4} \frac{x}{1+\cos 2x + \sin 2x} \, dx $.

The value of the integral: \[ I = \int_0^3 \frac{dx}{\sqrt{9 - x^2}} \] is:

Let $[t]$ denote the greatest integer less than or equal to t Then the value of the integral $\int\limits_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$ is equal to

If [t] denotes the greatest integer $≤ 1$, then the value of $3\frac{(e - 1)^2}{e}$ is:

The value of the integral $\int^2_1\bigg(\frac{t^4+1}{t^6+1}\bigg)$dt is

The value of the integral $∫_{\frac 12}^2 \frac{tan^{-1}x}{x}dx$ is equal to

Let the function f: [0,2] $\rightarrow$ R be defined as $f(x) = \begin{cases} e^{min}{x^2,x-[x]} & \quad x \in[0,1]\\ e^{[x-log_ex]} & \quad x\in[1,2] \end{cases}$
where [t] denotes the greatest integer less than or equal to t. Then the value of the integral $\int^2_0xf(x)dx$ is

If f : R $\rightarrow$ R be a continuous function satisfying $\int^{\frac{\pi}{2}}_0f(sin2x)sinxdx+\alpha\int^{\frac{\pi}{4}}_0f(cos2x)cosxdx=0$, then the value of $\alpha$ is

The value of $\frac{e^{-\frac{\pi}{4}}+\int^{\frac{\pi}{4}}_{0}e^{-x}\tan^{50}x\ dx}{\int^{\frac{\pi}{4}}_{0}e^{-x}(\tan^{49}x+\tan^{51}x)dx}$ is

The value of $\int\limits_0^π \frac {e^{cos⁡\ x} sin⁡x}{(1+cos^2⁡x)(e^{cos⁡\ x}+e^{−cos⁡\ x})}dx$ is equal to :

Let f be a real valued continuous function on [0, 1] and
$f(x) = x + \int_{0}^{1} (x - t) f(t) \,dt$
Then, which of the following points (x, y) lies on the curve y = f(x)?

If
$\int_{0}^{2} (\sqrt{2x} - \sqrt{2x - x^2}) \,dx = \int_{0}^{1} \left(1 - \sqrt{1 - y^2} - \frac{y^2}{2}\right) \,dy + \int_{1}^{2} (2 - \frac{y^2}{2}) \,dy + I$
then I equal is

Let
$M = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \\ \end{bmatrix}$
where α is a non-zero real number an
$N = \sum\limits_{k=1}^{49} M^{2k}. $ If $(I - M^2)N = -2I$
then the positive integral value of α is ____ .

The value of the integral
$\frac{48}{\pi^4} \int_{0}^{\pi}(\frac{3\pi x^2}{2} - x^3) \frac{ \sin(x)}{1 + \cos^2x} \, dx$
is equal to ________

The value of the integral $\begin{array}{l} \displaystyle\int\limits_0^{\frac{\pi}{2}}60\frac{\sin\left(6x\right)}{\sin x}dx \end{array}$ is equal to ______.

If
$n(2n+1)\int_{0}^{1}(1−xn)^{2n}dx=1177\int_{0}^{1}(1−x^n)^{2n+1}dx$
,then n ∈ N is equal to ______.