Definite Integration Questions - JEE

Suppose that F(x) is an antiderivative of f(x) = (sin x)/(x) − (3 sin 2x)/(1 + x²), where x > 0, then ∫ dx can be

∫₀² (ln x)/(x² + 1 + x²)dx is equal to

Evaluate $\int_3^5 x^5 \sqrt[3]{1 + 3x^4} \, dx$

Evaluate I = ∫02π [sin x + cos x] dx, where [·] is the greatest integer function.

Evaluate $\displaystyle \int_0^{\pi/2} \sin x \, dx$

If $f(x) = A \sin(x/2) + B$, $f'(\pi/2) = 2$, and $\sqrt{2}\int_0^1 f(x)dx = \frac{2A}{\pi}$, then the constants A and B are

If $f : \mathbb{R} \to \mathbb{R}$ is continuous and differentiable function such that $\int_0^x f(t) dt + \int_0^x t^3 dt = \int_0^x t^2 dt + f(1) \int_0^x f(t) dt - f(2) \int_0^x t dt + f(3)$, then the value of $f(4)$ is

Let $f(x) = \frac{x}{(1+nx^n)^{1/n}}$ for $n \ge 2$ and $g(x) = \underbrace{(f \circ f \circ \dots \circ f)}_{f \text{ occurs } n \text{ times}}(x)$. Then $\int x^{n-2} g(x) dx$ equals.

The value of $I = \int_0^{\pi/2} \dfrac{(\sin x + \cos x)^2}{\sqrt{1 + \sin 2x}}\, dx$ is

If $f\left(\dfrac{x-4}{x+2}\right) = 2x + 1$, $(x \in \mathbb{R} - \{1, -2\})$, then $\int f(x)\, dx$ is equal to (where $C$ is a constant of integration):

If $f(x) = \begin{vmatrix} \cos x & e^{x^2} & 2x\cos(x^2/2) \\ x^2 & \sec x & \sin x + x^3 \\ 1 & 2 & x + \tan x \end{vmatrix}$, the value of $\int_{-\pi/2}^{\pi/2} (x^2 + 1)[f(x) + f'(x)] dx$ is

If $I_n = \int \tan^n x\, dx$, then $I_4 + I_6 = a\tan^5 x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to:

Evaluate $\displaystyle\int_1^e(\ln x)^2\,dx$ [JEE Main 2019]

∫ln(x + √1 + x2) / √1 + x2 dx equals -

If ∫ 1 / ((x-1)^4 * (x+3)^6)^(1/10) dx = A * ((ax-1)/(bx+3))^B + C, where C is the constant of integration, then the value of α + β + 2AB is ________.

$\int \frac{x^2 + x}{(e^x + x + 1)^2} dx$ equals

If $f$ and $g$ are continuous functions in $[0, a]$ satisfying $f(x) = f(a-x)$ and $g(x) + g(a-x) = 2$, then prove that$\int_0^a f(x)\,g(x)\,dx = \int_0^a f(x)\,dx$.

Evaluate $\displaystyle\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{\sqrt{n(n+k)}}$ [JEE Main 2019]

Let F(x) be an indefinite integral of sin2x.Statement-1: The function F(x) satisfies F(x + π) = F(x) for all real x.becauseStatement-2: sin2(x + π) = sin2x for all real x.

∫\frac{x + x^{2/3} + x^{1/6}}{x(1 + x^{1/3})} dx = px^{2/3} + q \tan^{-1}(x^{1/6}) + c, \text{ then :-}

\int_{0}^{1} f'(1-t) e^{-\cos \pi t} dt - \int_{1}^{2} f'(2-t) e^{\cos \pi t} dt is equal to:

The number of solution(s) of the equation f(x) = x^3 in [0, 2\pi] be:

If In = ∫(sin x)n dx, n ∈ N, then 5I4 - 6I6 is equal to -

The value of: $\lim_{n \to \infty} \left( \frac{1}{n\sqrt{n+1}} + \frac{1}{n\sqrt{n+2}} + \frac{1}{n\sqrt{n+3}} + \ldots + \frac{1}{n\sqrt{2n}} \right)$ is:

The value of the integral $-\int_1^e \frac{\ln x}{x} dx$ is:

The integral ∫\frac{4x^5 - 7x^4 + 8x^3 - 2x^2 + 4x - 7}{x^2(x^2 + 1)^2} dx equals

The area bounded by the x-axis and the part of graph of y = cos x between x = $\frac{-\pi}{2}$ and x = $\frac{\pi}{2}$ is separated into two regions by the line x = k. If the area of the region for $\frac{-\pi}{2} \leq x \leq k$ is three times the area of the region for $k \leq x \leq \frac{\pi}{2}$, then k is equal to :

The value of ∫ ln(&frac{x-1}{x+1})⁄x2 - 1 dx is equal to

Evaluate: $\int x^x \left(\frac{(\ln x)^2}{x} + \frac{\ln x}{x} + \frac{1}{x}\right) dx$

If $I = \int \frac{x^2 - 1}{x^3(2x^4 - 2x^2 + 1)} dx$ is equal to:

18. $\int \frac{4x^5 - 7x^4 + 8x^3 - 2x^2 + 4x - 7}{x^2(x^2 + 1)^2} dx$ equals

∫\frac{x+x^{2/3}+x^{1/6}}{x(1+x^{1/3})}dx=px^{2/3}+q\tan^{-1}(x^{1/6})+c, \text{then :-}

∫ \frac{x^2 + x}{(e^x + x + 1)^2} dx equals

Primitive of $\frac{3x^4 - 1}{(x^4 + x + 1)^2}$ w.r.t. $x$ is -

9. $\int e^x \left( \frac{x^2 - 3}{(x - 1)^2} \right) dx$ is equal to -(where C is constant of integration)

[JEE Main 2021] If $\displaystyle\int f(x)\,dx=\psi(x)$, then $\displaystyle\int x^5 f(x^3)\,dx$ equals

If ∫ \frac{(x-1) dx}{x^2 \sqrt{2x^2 - 2x + 1}} = \frac{\sqrt{f(x)}}{g(x)} + C, where f(x) is a quadratic expression and g(x) is a monic linear expression.

14. $\int \sec^2 \theta(\sec \theta + \tan \theta)^2 d\theta$

The evaluation of $\int \frac{p x^{p+2q-1} - q x^{q-1}}{x^{2p+2q} + 2x^{p+q} + 1} dx$ is

$\displaystyle\int\frac{\sin(x-a)}{\sin(x+a)}\,dx$ equals

Value of $\int_1^5 \{x + 2\sqrt{x-1} + \sqrt{x-2}(x-1)\} dx$ is

If $f(x) = \int_0^x \frac{\sin t}{t} dt$, which of the following is true?

$\int \frac{\sec 5x}{\sin^3 x} dx$ equals to:

$\\int e^x \left( \frac{x^2 - 3}{(x - 1)^2} \right) dx$ is equal to -(where $C$ is constant of integration)

Let I(x) = ∫ \frac{x^2(\sec^2 x + \tan x)}{(\tan x + 1)^2} dx. If I(0) = 0 the I\left(\frac{\pi}{4}\right) is equal to

The integral ∫ sec^2 x / (sec x + tan x)^(9/2) dx equals (for some arbitrary constant K)

Evaluate $\displaystyle\int_0^{\pi/2}\frac{dx}{1+\tan^3 x}$ [JEE Main 2017]

Suppose J = ∫ \frac{\sin^2 x + \sin x}{1 + \sin x + \cos x} dx and K = ∫ \frac{\cos^2 x + \cos x}{1 + \sin x + \cos x} dx. If C is an arbitrary constant of integration then which of the following is/are correct?

Evaluate: $\int \frac{1 - x^2}{1 + x^4} dx$

If $I_n = \int_{0}^{\pi/2} \sin^n x \, dx$, then show that $I_n = \left(\frac{n-1}{n}\right) I_{n-2}$

Definite Integration Questions (1340)