JEE Definite Integration PYQs With Video Solutions PDF

Let $$f$$ be a twice differentiable function on $$\mathbb{R}$$. If $$f'(0) = 4$$ and $$f(x) + \displaystyle\int_0^x (x-t)f'(t) dt = (e^{2x} + e^{-2x})\cos 2x + \dfrac{2}{a}x$$, then $$(2a+1)^5a^2$$ is equal to ______.

Let f be a continuous function satisfying $$\int_0^{t^2} f(x) + x^2 dx = \frac{4}{3}t^3$$, $$\forall t > 0$$. Then $$f\left(\frac{\pi^{2}}{4}\right)$$ is equal to

The function $$f(x)$$, that satisfies the condition $$f(x) = x + \int_0^{\pi/2} \sin x \cos y f(y) dy$$, is:

Suppose $$𝑦 = 𝑦𝑥$$ be the solution curve to the differential equation $$\frac{dy}{dx}-y=2-e^{-x}$$ such that $$\lim_{x \rightarrow \infty} yx$$ If $$𝑎$$ and $$𝑏$$ are respectively the $$𝑥 -$$ and $$𝑦 -$$ intercept of the tangent to the curve at $$𝑥 = 0$$, then the value of $$𝑎 - 4𝑏$$ is equal to _______.

$$\displaystyle\int_0^{20\pi} (|\sin x| + |\cos x|)^2 dx$$ is equal to:

Let $$\left[\cdot\right]$$ be the greatest integer function. If $$(\alpha = \int_{0}^{64} \left( x^{1/3} - [x^{1/3}] \right)\, dx $$, then $$\frac{1}{\pi} \int_{0}^{\alpha\pi } \left( \frac{\sin^{2}\theta } {\sin^{6}\theta + \cos^{6}\theta} \right) d\theta$$ is equal to ____ .

The value of $$\int_{\frac{-\pi}{6}}^{\frac{\pi}{6}}\left(\frac{\pi+4x^{11}}{1-\sin(|x|+\frac{\pi}{6})}\right)dx$$ is equal to :

Let $$g(x) = \int_0^x f(t)dt$$, where $$f$$ is continuous function in $$[0, 3]$$ such that $$\frac{1}{3} \le f(t) \le 1$$ for all $$t \in [0, 1]$$ and $$0 \le f(t) \le \frac{1}{2}$$ for all $$t \in (1, 3]$$.
The largest possible interval in which $$g(3)$$ lies is :

Let $$f(x)$$ be a function satisfying $$f(x) + f(\pi - x) = \pi^2$$, $$\forall x \in \mathbb{R}$$. Then $$\int_0^\pi f(x) \sin x \, dx$$ is equal to

A value of $$\alpha$$ such that $$\int_\alpha^{\alpha+1} \frac{dx}{(x + \alpha)(x + \alpha + 1)} = \log_e\left(\frac{9}{8}\right)$$ is

The area of the region enclosed by the parabola $$y = 4x - x^2$$ and $$3y = (x - 4)^2$$ is equal to

The area (in sq. unit) bounded by the curve $$4y^2 = x^2(4-x)(x-2)$$ is equal to

Let $$f(x)$$ be a differentiable function defined on $$[0, 2]$$ such that $$f'(x) = f'(2 - x)$$ for all $$x \in (0, 2)$$, $$f(0) = 1$$ and $$f(2) = e^2$$. Then the value of $$\int_0^2 f(x)dx$$ is

The area of the region $$\{(x,y): y \leq \pi - |x|, \; y \leq |x \sin x|, \; y \geq 0\}$$ is :

Let $$\int_{-2}^{2} (|\sin x| + [x \sin x])\,dx = 2(3 - \cos 2) + \beta$$, where $$[\cdot]$$ is the greatest integer function. Then $$\beta \sin\left(\frac{\beta}{2}\right)$$ equals :

If the integral $$\int_0^{10} \frac{|\sin 2\pi x|}{e^{[x]}}dx = \alpha e^{-1} + \beta e^{-\frac{1}{2}} + \gamma$$, where $$\alpha, \beta, \gamma$$ are integers and $$[x]$$ denotes the greatest integer less than or equal to $$x$$, then the value of $$\alpha + \beta + \gamma$$ is equal to:

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 __________.

Let $$[\cdot]$$ denote the greatest integer function. Then the value of $$\int_0^3 \left(\frac{e^x + e^{-x}}{[x]!}\right) dx$$ is :

Let $$f : [0, 1] \to [0, 1]$$ be the function defined by $$f(x) = \frac{x^3}{3} - x^2 + \frac{5}{9}x + \frac{17}{36}$$. Consider the square region $$S = [0, 1] \times [0, 1]$$. Let $$G = \{(x, y) \in S : y > f(x)\}$$ be called the green region and $$R = \{(x, y) \in S : y < f(x)\}$$ be called the red region. Let $$L_h = \{(x, h) \in S : x \in [0, 1]\}$$ be the horizontal line drawn at a height $$h \in [0, 1]$$. Then which of the following statements is(are) true?

Let $$A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix}$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $$\alpha A^2 + \beta A = 2I$$. Then $$\alpha + \beta$$ is equal to

Let $$\vec{a} = 2\hat{i} - \hat{j} + 5\hat{k}$$ and $$\vec{b} = \alpha \hat{i} + \beta \hat{j} + 2\hat{k}$$. If $$\left(\left(\vec{a} \times \vec{b}\right) \times \hat{i}\right) \cdot \hat{k} = \frac{23}{2}$$, then $$\left|\vec{b} \times 2\hat{j}\right|$$ is equal to

The minimum value of the twice differentiable function $$f(x) = \int_0^x e^{x-t} f'(t) dt - (x^2 - x + 1)e^x$$, $$x \in \mathbb{R}$$, is

The area of the region enclosed by the parabolas $$y = x^2 - 5x$$ and $$y = 7x - x^2$$ is ______

If $$I_n = \int_{\pi/4}^{\pi/2} \cot^n x \, dx$$, then

The value of the integral, $$\int_1^3 [x^2 - 2x - 2] dx$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is

If $$\int_{-a}^{a} (|x| + |x - 2|) dx = 22$$, $$a > 2$$ , then $$\int_{-a}^{a} (x + |x|) dx$$ is equal to ______

Let P be the foot of the perpendicular from the point Q(10,-3,-1) on the line $$\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z+1}{-2}$$. Then the area of the right angled triangle PQR , where R is the point (3,-2,1),is

The integral $$\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$$ is equal to

The integral $$\int \frac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3} dx$$, is equal to

If $$\int_0^{\pi/3} \frac{\tan\theta}{\sqrt{2k\sec\theta}} d\theta = 1 - \frac{1}{\sqrt{2}}$$, $$(k \gt 0)$$, then the value of $$k$$ is: