Definite Integration Questions - JEE

Let F(x) be a non-negative continuous function defined on ℝ such that F(x) + F(x − 1/2) = 3.Find the value of ∫₀¹⁵⁰⁰ F(x)dx.

Consider f(x) = \frac{x^2}{1+x^3}; g(t) = \int f(t)dt. If g(1) = 0 then g(x) equals -

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

$\int_{0}^{1} \cot^{-1}(1-x+x^2) dx$ equals -

Let $f(x)=x+\dfrac{a}{\pi^2-4}\sin x+\dfrac{b}{\pi^2-4}\cos x$, $x\in\mathbb{R}$, be a function which satisfies $f(x)=x+\displaystyle\int_0^{\pi/2}\sin(x+y)f(y)\,dy$. Then $(a+b)$ is equal to:

The value of $ a_n = \dfrac{1}{\pi} \int_{-\pi}^{\pi} (x + x^2) \cos nx \, dx $ is:

If f : ℝ → ℝ is a continuous and differentiable function such that ∫₀ˣ f(t)dt − f(1)∫₀ˣ t dt = 1 and ∫₀² t³ dt − f(2)∫₀ˣ t² dt, then the value of f(4) is

Given $ I = \int_0^{\pi/2} \dfrac{\sin^3 x}{\sin x + \cos x}\,dx $. Find the value of $I$.

56. The value of $\displaystyle\lim_{n \to \infty} \sum_{r=1}^{n} \frac{\pi}{n} \cdot \frac{1}{\sin\!\left(\dfrac{(n+r)\pi}{4n}\right)}$ is equal to:

If $f(x) = \begin{cases} 3[x]-5\frac{|x|}{x}, & x \neq 0 \\ 2, & x = 0 \end{cases}$ then $\int_{-3/2}^{2} f(x) dx$ is equal to ([.] denotes the greatest integer function)

$\int_2^3 \frac{2x^2}{x^4 + 3x^2 + 1} dx$ is equal to

Let f be a differentiable function from R to R such that $|f(x) - f(y)| \leq 2|x - y|^{3/2}$, for all $x, y \in R$. If $f(0) = 1$ then \[\int_0^1 f^2(x)\,dx\] is equal to:

The value of $\int_{-\pi/2}^{\pi/2} (q\sin^3 x + r\sin^5 x + s\sin^5 x)\,dx$ depends on

If \theta_1 and \theta_2 be respectively the smallest and the largest values of \theta in (0, 2\pi) - \{\pi/2\} which satisfy the equation 2\cot\theta - \frac{5}{\sin\theta} + 4 = 0, then \int_{\theta_1}^{\theta_2} \cos^2 3\theta d\theta equals

Statement-1: The value of the integral $\int_{\pi/6}^{\pi/3} \dfrac{dx}{1 + \sqrt{\tan x}}$ is equal to $\dfrac{\pi}{6}$.Statement-2: $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a+b-x) \, dx$.

If ∫_{sin x}^1 t² f(t) dt = 1 − sin x for all x ∈ (0, π/2), then f(1/3) is:

Let f and g be continuous functions on [0, a] such that f(x) = f(a - x) and g(x) + g(a - x) = 4, then ∫a0 f(x)g(x) dx is equal to

The value of $\int_0^{\pi/2} \frac{\cos^2 x}{1 + \cos x} dx$, where $[\cdot]$ denotes greatest integer function, is

Evaluate \[I = \int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx\]

Evaluate $\int_1^2 [x^3 - 1] \, dx$, where $[\cdot]$ denotes the greatest integer function.

The value of $\int_{-1}^{1} \frac{1+\sin x + x^2}{1+\sin x + x^2} dx$ is

The integral $\int_{1/2}^{1/2} \frac{[x]}{1-x} dx$ equals

Let $I_n = \int_{1/(n+1)}^{1/n} \tan^{-1}(nx) \, dx$, then $\lim_{n \to \infty} n^2 I_n$ is equal to

If $f''(x) = k$ in $[0, a]$, then $\int_0^a \frac{f(x) - \frac{x^2}{2!} f''(x) - \frac{x^3}{3} xf(x)}{...} dx$ is

The value of ∫₋₁⁵ sgn({x}) dx, where {⋅} denotes the fractional part function, is:

If f(x + y) = f(x) + f(y) for all x and y, and \int_a^{a+1} (x-1)^2 f(x-1) \, dx = 2 and \int_1^3 (x-1)^2 f(x-1) \, dx = b, then a is equal to

Evaluate $\int_{0}^{\pi} \frac{dx}{1 + 3 \cos^2 x}$.

Let f be integrable over [0, a] for any real a. If we defineI1 = ∫₀^(π/2) cos θ f(sin θ + cos θ) dθand I2 = ∫₀^(π/2) sin² θ f(sin θ + cos² θ) dθthen the relationship between I1 and I2 is:

If $f : [0, \pi] \to \mathbb{R}$ is continuous and $\int_0^\pi f(x) \sin x \, dx = \int_0^\pi f(x) \cos x \, dx = 0$, then

Let $2\int_{1}^{4} x(5 - f^{-1}(x))\,dx$ equals (given that $g(3) = 2$ and other relevant conditions hold). Find the value of $2\int_{1}^{4} x(5 - f^{-1}(x))\,dx$.

For 0 \leq x \leq \frac{\pi}{2}, \int_{1/2}^{\pi/2} \cos x \, d(\cos x) is

Let $I = \int_0^1 \frac{\sin x}{\sqrt{x}}\, dx$ and $J = \int_0^1 \frac{\cos x}{\sqrt{x}}\, dx$. Then which one of the following is true?

If $F(x) = \int_{x}^{x^2} \sqrt{\tan t} dt$, then find $F'(x)$.

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

The integral \int_{\pi/6}^{\pi/4} \frac{dx}{\sin^2 x(\tan^5 x + \cot^5 x)} is equal to

The integral ∫√cot x e^√sin x √cos x dx equals

∫ 3x² + 1 / (x² - 1)³ dx equals (where K is constant of integration)

Let a function h(x) be defined as h(x) = 0, for all x ≤ 0. Also ∫h(x).f(x)dx = f(0) for every function f(x). Then the value of the definite integral ∫₋₁⁰ h'(x).sin x dx is

Evaluate $\int_{0}^{2\pi}(\sin x + |\sin x|)\,dx$

Consider $0 \leq \int_0^1 (f'(x)-1)^2\, dx$. If $f(0)=0$ and $f(1)=1$, then find the minimum value of $\int_0^1 (f'(x))^2\, dx$.

The value of $\int_0^{\pi/2} \frac{dx}{1+\tan^3 x}$ is

Let f be a non-negative function defined on the interval [0, 1]. If $\int_0^x [f'(t)]^2 dt \leq \int_0^x f(t) dt$ for all $0 \leq x \leq 1$ and $f(0) = 0$, then

Show $\dfrac{7}{10}\le\displaystyle\int_1^2\frac{x^2+1}{x^3+1}\,dx\le\dfrac{7}{5}$. Evaluate the integral. [JEE Main 2021]

∫₋π/₃^0 cot⁻¹(cot 2cos x − 1/(2cos x − 1))dx is equal to

Let g(x) be an antiderivative for f(x). Then ln(1+(g(x))^2) is an antiderivative for

Let $f$ be a positive function. Let $I_1 = \int_{1-k}^{k} x f\{x(1-x)\}dx$ and $I_2 = \int_{1-k}^{k} f\{x(1-x)\}dx$, where $2k-1 > 0$. Then $I_1/I_2$ is equal to

∫₀⁴ ((y² + 4y + 5)sin(y−2))/(2y² + 8y + 1)dy is equal to

Let $J_n = \displaystyle\int_0^{\pi/2} (1 - \sin x)^n \sin 2x\, dx$. Find $\displaystyle\sum_{n=0}^{\infty} J_n$.

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

∫₀ᵖ e^(sin 2x) cos 3x dx is equal to

Definite Integration Questions (1340)