Definite Integration Questions - JEE

If for a real number y, [y] is the greatest integer less than or equal to y, then the value of the integral $\int_0^{3\pi/2} [2\sin x] dx$ is

[JEE Main 2023] $\displaystyle\int\frac{3\sin x+2\cos x}{5\sin 2x+3}\,dx$ equals (where $C$ is constant)

If $[\cdot]$ is G.I.F., then $\int_0^{10} \frac{[x^2]}{4[x^2 - 28x + 196] + [x^2]} dx$ is

The value of $\int_0^{\pi} [2\sin x] dx$ where [ ] represents the greatest integer function is

Let T > 0 be a fixed real number. Suppose f is a continuous function such that for all x ∈ ℝ, f(x + T) = f(x). If $I = \int_0^T f(x) \, dx$, then the value of $\int_0^{3T} f(2x) \, dx$ is

$\int \sin^2(\ln x) \, dx$ is equal to

If $I(m, n) = \int_0^1 t^m (1+t)^n dt$, then the expression for $I(m, n)$ in terms of $I(m+1, n-1)$ is

If the function $\int_0^x f(t) dt = 5$ as $|x| \to 1$, then the value of 'a' so that the equation $2x + \int_0^x f(t) dt = a$ has atleast two roots of opposite signs in $(-1, 1)$ is

Let $f(x)$ be a periodic function with period 3 and $\int_0^3 f(t) \, dt = 7$. If $g(x) = \int_0^x f(t+n) \, dt$ where $n = 3k$, $k \in \mathbb{N}$, then

If $f(x) = \displaystyle\int_2^x \frac{dt}{1+t^4}$, then:

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

890. Let $L = \lim_{n \to \infty} \dfrac{1}{n^3} \sum_{k=1}^{n} k^2 e^{\frac{k}{n}}$, then find the value of $e - L$.

The value of \int_{\pi/4}^{\pi/2} \sin 2x \tan^{-1}(\sin x) \, dx is equal to

If $f(x) = \displaystyle\int_2^x \frac{dt}{1+t^4}$, then:

Let f(x) = \frac{2\sin^2 x - 1}{\cos x} + \frac{\cos x(2\sin x + 1)}{1 + \sin x} then \int e^x(f(x) + f'(x))dx (where c is the constant of integration)

For any x ∈ ℝ, and f be a continuous function. Let I1 = ∫sin²x1+cos2x tf(t(2-t)) dt, I2 = ∫sin²x1+cos2x f(t(2-t)) dt, then I1 = ?

If $I_1 = \int_0^1 \frac{1+x^8}{1+x^4} dx$ and $I_2 = \int_0^1 \frac{1+x^9}{1+x^3} dx$, then:

Let f(x) = \int_x^2 \frac{1}{1+y^3} dy. The value of the integral \int_0^2 xf(x) dx is equal to:

Which of the following is true for $x > 0$? Where $g(x) = \int_0^x (3t^2 + 2t + 9) dt$, $f(x)$ is a decreasing function, $AB = f(x)i + g(x)j$, and $AC = g(x)i + f(x)j$ are the two smallest sides of triangle ABC whose circumcentre lies outside the triangle.

Evaluate \[\int \sqrt{\frac{\cos^{\frac{7}{2}} x}{\sin^{\frac{11}{2}} x}}\, dx\]

∫ sin-1√xx+a dx equals

∫ \frac{x^2(1-\ln x)}{\ln^4 x - x^4} dx equals

$\displaystyle\int\frac{4x^3-7x^2+8x-4}{x^2(1+x^2)}\,dx$ equals (where $C$ is the constant of integration)

Let I = ∫ \frac{e^x}{e^{4x} + e^{2x} + 1} dx, J = ∫ \frac{e^{-x}}{e^{-4x} + e^{-2x} + 1} dx. Then, for an arbitrary constant c, the value of J - I equals

Evaluate: \ 6\int \frac{dx}{x^4 + x^2}

The value of $\int_0^{\infty} \frac{54a}{(3 + a + 4y)^4}\, dy$ is ______.

Evaluate the following: 19. $\int_0^1 \frac{\sin^2 x - \cos^2 x}{(\sin^3 x + \cos^3 x)^2} \, dx$

Evaluate the following: 17. $\int_0^{\pi/5} \cos^2\frac{x}{2} \cdot \sin x \, dx$

Evaluate $\int \dfrac{2x+3}{\sqrt{1+x+x^2}}\, dx$.

If $f(1+x) = f(1-x)$ ($\forall x \in \mathbb{R}$), then the value of the integral $I = \int_{-\infty}^{\infty} \frac{f(x)}{1+2^{(x-a)}} dx$ is

Let $I_1 = \int_0^1 \frac{x}{x^2+1} dx$ and $I_2 = \int_1^\infty \frac{x}{x^2+1} dx$, then

The integral $\int \frac{dx}{(x + 4)^{8/7}(x - 3)^{6/7}}$ is equal to (where C is a constant of integration)

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

Prove that $\int \sqrt{x^2 - a^2} dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log |x + \sqrt{x^2 - a^2}| + C$

If $f(x), g(x), h(x)$ and $f'(x)$ are polynomials in $x$, and$\left(\int_1^x f(x)h(x) dx\right) \left(\int_1^x g(x)f'(x) dx\right) - \left(\int_1^x f(x)f'(x) dx\right) \left(\int_1^x g(x)h(x) dx\right)$is divisible by $(x-1)^l$, find the maximum value of $l$.

Let $g(x) = x^c e^{2x}$ & let $f(x) = \int_0^x e^{2t}(3t^2+1)^{1/2} dt$. For a certain value of 'c', the limit of $\frac{f'(x)}{g'(x)}$ as $x \to \infty$ is finite and non-zero, then:

For $x \in (0, 1)$ arrange $f_1(x) = \frac{1}{9-x^2}, f_2(x) = \frac{1}{9-2x^2}$ and $f_3(x) = \frac{1}{9-x^2-x^3}$ in ascending order and hence prove that $$\frac{1}{6} \ln 2 < \int_{0}^{1} \frac{1}{9-x^2-x^3} dx < \frac{1}{6\sqrt{2}} \ln 5$$

The value of $k\in\mathbb{N}$ for which the integral $I_n=\int_0^1(1-x^k)^n\,dx$, $n\in\mathbb{N}$, satisfies $147I_{20}=148I_{21}$ is:

\[\left|2\left(x^2 + \frac{1}{x^2}\right) + |1 - x^2|\right| = 4\left(\frac{3}{2} - 2^{x^2 - 3} - \frac{1}{2^{x^2+1}}\right)\]If $x_1$ and $x_2$, where \(x_1 [Note: |·| denotes the absolute value function, {·} denotes the fraction part function, [·] denotes the floor function]

Evaluate: $\int_{0}^{\pi/2} \dfrac{\sin 8x \log(\cot x)}{\cos 2x}\, dx$.

If $\displaystyle\int_0^1 f(x)\,dx = 1 + 2\int_0^1 x\,f(x)\,dx$, and $f(x)=1+cx$, find $c$. [JEE Main 2021]

If ∫ (x-1) dxx22x2-2x+1 = f(x)g(x) + C, where f(x) is a quadratic expression and g(x) is a monic linear expression.

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

If $f\left(\frac{1-x}{1+x}\right)=x$ and $g(x)=\int f(x)dx$ then(A) $g(x)$ is continuous in domain(B) $g(x)$ is discontinuous at two points in its domain(C) $\lim_{x\to\infty} g'(x)=-1$(D) $\int g(x)dx=-\frac{x^2}{2}+(2x+1)\ln\left(\frac{1+x}{e}\right)+C$

The value of $\sum_{n=1}^{100} \int_{n-1}^{n} e^{x-[x]} dx$, where $[x]$ is the greatest integer $\leq x$, is (JEE Main 2021)

Let $A_n = \int \tan^n x \, dx$, $\forall n \in \mathbb{N}$. If $A_{n} + A_{n-2} = \frac{\tan^{n-1} x}{n-1} + \lambda$ (where $\lambda$ is an arbitrary constant), then the value of $m$ is equal to

The value of the integral $\int e^{x+1}(2x^2 - \frac{1}{x} + 1) dx$ is equal to (where $C$ is the constant of integration)

If $\int \frac{2\sin x}{\ln(1 + (f(x)) + C}$ (where $x > 0$ and $C$ is the constant of integration) then the range of $f(x)$ is

$\int e^{x\left(1 + n \cdot x^{n-1} + x^{2n}\right)} \frac{1}{(1+x^n)\sqrt{1+x^{2n}}} dx$ is equal to

Definite Integration Questions (1340)