Definite Integration Questions - JEE

Let f(x) be a continuous function in (0, 1) satisfying $\int_0^1 x\sqrt{x}\, f(x)(1 - \sqrt{x}f(x))\, dx = \dfrac{1}{8}$. Number of solutions of the equation $f(x) = e^x$ is:

Let $f(x)=\displaystyle\int\frac{(2-x^2)\cdot e^x}{(\sqrt{1+x})(1-x)^{3/2}}\,dx$. If $f(0)=0$, then $f\!\left(\dfrac{1}{2}\right)$ is equal to:

$\int (\tan x + \cot x) dx$ is equal to

$\int_1^2 \sqrt{\frac{2+x}{2-x}}\, dx$ is equal to

If $\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\dfrac{e^{\frac{k}{n}}+e^{-\frac{k}{n}}}{n\sqrt{1-e^{\frac{2k}{n}}-e^{-\frac{2k}{n}}}} = \sin^{-1}\!\left(\dfrac{e^a - e^{-a}}{b}\right)$ where $a$ and $b$ are positive integers, then the value of $a+b$ is:

If ∫√(sec 2x - 1) dx = α log_e |cos 2x + β + √(cos 2x (1 + cos 2x / β))| + constant, then β - α is equal to ____.

Let $f(t)=\displaystyle\int\left(\frac{1-\sin(\log_e t)}{1-\cos(\log_e t)}\right)dt$, $t>1$. If $f(e^{\pi/2})=-e^{\pi/2}$ and $f(e^{\pi/4})=\alpha e^{\pi/4}$, then $\alpha$ equals

If $\int \frac{1+3\tan x(\tan x + \sec x)}{\tan x} dx = a \log \left|\cos \frac{x}{2} + \sin \frac{x}{2}\right| + C$ where $0 , then \(a$ is equal to

Given $f(x) = \int \dfrac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} dx$. If $f(0) = 0$, then find the value of $f(1)$.

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

If $\int\left[\ln\left(\frac{\cos 2\theta}{1+\sin 2\theta}\right) + \ln\left(\frac{1+\sin 2\theta}{1-\sin 2\theta}\right)^{\cos^2\theta}\right]d\theta$ is equal to $\frac{1}{a}\sin 2\theta\ln\left|\frac{\cos\theta + \sin\theta}{\cos\theta - \sin\theta}\right| + b\ln|\cos 2\theta| + c$ where $a, b \in \mathbb{R} - \{0\}$ & $c$ is integration constant such that $\cos\theta > \sin\theta > 0$ then $(a+b)$ is

$\int \tan x dx$ is equal to

Let $f$ & $g$ be differentiable function for all $x \in \mathbb{R}$ & have the following properties (i) $f'(x) = f(x) - g(x)$ (ii) $g'(x) = g(x) - f(x)$ (iii) $f(0) = 5$ (iv) $g(0) = 1$ Then the value of $|f(\ln 2) + g(\ln 3)|$ is equal to

If $\int\frac{x^3 + x + 1}{x^4 + x^2 + 1}dx = A_1\ln(x^2 + x + 1) + A_2\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right) + A_3\tan^{-1}\left(\frac{2x-1}{\sqrt{3}}\right) + A_4\tan^{-1}\left(\frac{2x^2+1}{\sqrt{3}}\right) + c$ then the value of $(A_1 + A_2 + A_3 + A_4)$ is

Let $\int\frac{\ln\left(x + \sqrt{1+x^2}\right)}{\sqrt{1+x^2}}dx = fog(x) + c$, where $f(x) = \frac{x^2}{2}$ and $g$ are some functions and $c$ is an arbitrary constant. If $\int f(x)g(x)dx = ax^3g(x) + b\left(1+x^2\right)^{3/2} + c\left(1+x^2\right)^{1/2} + d$, then $\left(\frac{1}{a+b+c}\right)$ is equal to

If $\int\frac{(\cos x - \sin x + 1 - x)}{e^x + \sin x + x}dx = \ln(f(x)) + g(x) + c$ where $c$ is the constant of integration & $f(x)$ is positive, then $\frac{f(x) + g(x)}{e^x + \sin x}$ is

If $\int\left(x^{2010} + x^{804} + x^{402}\right)\left(2x^{1008} + 5x^{402} + 10\right)^{10a}dx = \frac{1}{10a}\left(2x^{2010} + 5x^{804} + 10x^{402}\right)^{10a} + c$, where $c$ is constant then $a$ is equal to

Comment upon the nature of roots of the quadratic equation $x^2 + 2x + k = \int_0^k |1+k| dr$ depending on the value of $k \in \mathbb{R}$.

Let $f: R \to R$ be a differentiable function and $f(1) = 4$. Then the value of $\lim_{x \to 1} \int_4^{f(x)} \frac{2t}{x-1} dt$ is

The value of the integral $\displaystyle\int_{-2}^{2} \frac{\sin^2 x}{\left[\dfrac{x}{\pi}\right]+\dfrac{1}{2}}\, dx$ (where $[x]$ denotes the greatest integer less than or equal to $x$) is:

The integral ∫ sec2 x / (sec x + tan x)9/2 dx equals (for some arbitrary constant K)

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

If $\int \dfrac{3\tan\!\left(x - \dfrac{\pi}{4}\right)}{\cos^2 x\,\sqrt{\tan^3 x + \tan^2 x + \tan x}}\, dx = k\tan^{-1}\!\left(\sqrt{\tan x + 1 + \cot x}\right) + C$, then the value of $k$ is: [where $C$ is constant of integration.]

Let $a, b, c$ be non-zero real numbers such that $\int_0^1 (1 + \sin^8 x)(ax^2 + bx + c)\,dx = \int_0^2 (1 + \sin^8 x)(ax^2 + bx + c)\,dx = 0$. Then the equation $ax^2 + bx + c = 0$ has

If $\displaystyle\int x^{26}(x-1)^{17}(5x-3)\, dx = \dfrac{x^{27}(x-1)^{18}}{k} + C$, where $C$ is constant of integration, then the value of $k$ is:

MATRIX MATCH TYPE QUESTIONList-IList-II(P) $\int \frac{x^2(x^6+x^5-1)dx}{(2x^6+3x^5+2)^2}$(1) $-\frac{1}{3} \frac{1}{(x^3-x+1)} + C$(Q) $\int \frac{(x^5+x^4+x^2)dx}{\sqrt{4x^7+5x^6+10x^4}}$(2) $\frac{1}{2}(1+x^{-2}+x^{-5})^{-2} + C$(R) $\int \frac{(2x^{12}+5x^9)dx}{(x^5+x^3+1)^3}$(3) $-\frac{1}{6}(2x^3+3x^2+2x^{-3})^{-1} + C$(S) $\int \frac{x^2-\frac{1}{3}}{(x^3-x+1)^2} dx$(4) $x\left(\frac{x^3}{25} + \frac{x^2}{20} + \frac{1}{10}\right)^{\frac{1}{2}} + C$

If $\int \frac{dx}{\cos^3 x \cdot \sqrt{2\sin 2x}} = (\tan x)^A + C(\tan x)^B + k$, where $k$ is a constant of integration, then $A + B + C$ equals

If $\int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \ln |9e^{2x} - 4| + C$, then

Let $f:[0,5] \to \mathbb{R}$ be such that $f''(x) = f''(5-x)$, $\forall x \in [0,5]$, $f'(0) = 1$ and $f'(5) = 7$, then the value of $\int_1^4 f'(x)\, dx$ is:

Let $f(x^3 + 1) = t$. If $\displaystyle\int_2^{10} g(t)\,dt = 24$, then find the value of $\displaystyle\int_4^{20} g^{-1}(x)\,dx$.

Let I = ∫ e^x / (e^4x + e^2x + 1) dx, J = ∫ e^-x / (e^-4x + e^-2x + 1) dx. Then, for an arbitrary constant c, the value of J - I equals

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.

15. $\int [\sin \alpha \sin(x - \alpha) + \sin^2(\frac{x}{2} - \alpha)] dx$ equals

Let $I_n = \int \tan^n x\, dx$, $(n > 1)$. If $I_4 + I_6 = a\tan^5 x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to

The value of $\displaystyle\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx$ is ______ (up to four decimal places).

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

If $\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} dx = \frac{1}{12} \tan^{-1}(3 \tan x) + \text{constant}$, then the maximum value of $a \sin x + b \cos x$, is :

∫dx(x-α)√((x-α)(x-β))equals

Let I(x) = ∫ 6 / (sin² x (1 - cot x)²) dx. If I(0) = 3, then I(π/12) is equal to:

The integral ∫ sec²x / (sec x + tan x)^9/2 dx equals (for some arbitrary constant K)

The evaluation of ∫ p xp+2q-1 - q xq-1 ⁄ x2p+2q + 2xp+q + 1 dx is

$\int (g(x) + 1) \sin x$ dx is equal to

If $f(x)$ is continuous and $n \in \mathbb{N}$ then the value of $\int_{-1}^{1} [f(x) - f(-x)]^{2n+1}\,dx$ is

If $I_n = \int \cot^n x \, dx$ (where $u = \cot x$), then the value of $I_2 + I_3 + I_4 + \ldots + I_9 + I_{10}$ is $l$

The integral $\int \sqrt{1 + 2\cot x(\csc x + \cot x)}\, dx$ is equal to (where $C$ is a constant of integration):

MATRIX MATCH TYPE QUESTIONList-IList-II(P) $\int \frac{x^2(x^6+x^5-1)dx}{(2x^6+3x^5+2)^2}$(1) $-\frac{1}{3}\frac{1}{(x^3-x+1)}+C$(Q) $\int \frac{(x^5+x^4+x^2)dx}{\sqrt{4x^7+5x^6+10x^4}}$(2) $\frac{1}{2}(1+x^{-2}+x^{-5})^{-2}+C$(R) $\int \frac{(2x^{12}+5x^9)dx}{(x^5+x^3+1)^3}$(3) $-\frac{1}{6}(2x^3+3x^2+2x^{-3})^{-1}+C$(S) $\int \frac{x^2-\frac{1}{3}}{(x^3-x+1)^2}dx$(4) $x\left(\frac{x^3}{25}+\frac{x^2}{20}+\frac{1}{10}\right)^{\frac{1}{2}}+C$(where C is the constant of integration.)

Let $F(x) = f(x) + f\!\left(\dfrac{1}{x}\right)$, where $f(x) = \int_{1}^{x} \dfrac{\log t}{1+t} \, dt$. Then $F(e)$ equals

Evaluate $\int \frac{\cos x + x \sin x}{x(x + \cos x)} dx$

Prove that $\displaystyle\int_0^\infty \frac{dx}{1+x^4} = \int_0^\infty \frac{x^2\,dx}{1+x^4} = \frac{\pi}{2\sqrt{2}}$.

If $g(x) = \int_0^x \cos 4t\, dt$, then $g(x+\pi)$ equals

Definite Integration Questions (1340)