Definite Integration Questions - JEE

Let $r_k=\dfrac{\int_0^1(1-x^7)^k\,dx}{\int_0^1(1-x^7)^{k+1}\,dx}$, $k\in\mathbb{N}$. Then the value of $\displaystyle\sum_{k=1}^{10}\dfrac{1}{7(r_k-1)}$ is equal to ________.

If $f\left(\dfrac{3x-4}{3x+4}\right) = x + 2$, $x \neq -\dfrac{4}{3}$, and $\int f(x)\,dx = A\log|1-x| + Bx + C$, then the ordered pair $(A, B)$ is equal to (where $C$ is a constant of integration)

Let $[t]$ denote the largest integer $\leq t$. If $\int_0^3\left([x^2]+\left[\dfrac{x^2}{2}\right]\right)dx=a+b\sqrt{2}-\sqrt{3}-\sqrt{5}+c\sqrt{6}-\sqrt{7}$, where $a,b,c\in\mathbb{Z}$, then $a+b+c$ is equal to ________.

The integral $\int_0^{\pi/4}\dfrac{136\sin x}{3\sin x+5\cos x}\,dx$ is equal to:

The value of the integral $\int_{-1}^{2}\log_e\left(x+\sqrt{x^2+1}\right)dx$ is:

Let I(x) = \int 11 dx 15 . If I(37) - I(24) = 1 4 ( 1 1 - 1 1 ) , b, c \in N , then 3( b + c) is equal to (x-11) 13 (x+15) 13 b 13 c 13

The integral 80 \int 4 sin \theta+cos \theta ) ( 9+16 d\theta is equal to : 0 sin 2\theta

If I(m, n) = \int 1 0 x m-1 (1 - x) n-1 dx, m, n > 0 , then I (9, 14) + I (10, 13) is

If f (x) = \int 1/4 1 1/4 dx, f (0) = -6 , then f (1) is equal to : x (1+x )

Let for some function y = f (x), \int and f (2) = 3. Then f (6) is equal to x 2 tf (t)dt = x f (x), x > 0 0

$\displaystyle\lim_{n\to\infty} \frac{n^2}{\left((n^2+1^2)(n^2+2^2)\cdots(n^2+n^2)\right)^{\frac{1}{n}}}$ equals:

The value of $\displaystyle\sum_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int_0^r x|\sin\pi x|\,dx\right)}\right|\right)$ is _____

If $\displaystyle\sum_{i=1}^{n}(\sin^{-1} x_i + \cos^{-1} y_i) = \frac{9\pi}{...}$, evaluate $\displaystyle\int_{-1}^{1} x\ln(1+x^2)\cdot\frac{e^x}{1+e^{2x}}\,dx$

273. If $I_1 = \int_0^1 \frac{x^{7/2}(1-x)^{9/2}}{30}\, dx$ and $I_2 = \int_0^1 \frac{x^{7/2}(1-x)^{9/2}}{(x+5)^{10}}\, dx$ and $\frac{I_1}{I_2} = 5a^3\sqrt{a}$, where $a \in N$, then the value of $a$ is:

If $\displaystyle\int\dfrac{\sin^{3/2}x+\cos^{3/2}x}{\sqrt{\sin^3x\cos^3x\sin(x-\theta)}}\,dx=A\sqrt{\cos\theta\tan x-\sin\theta}+B\sqrt{\cos\theta-\sin\theta\cot x}+C$, where $C$ is the integration constant, then $AB$ is equal to

31. The value of $\lim_{h \to 0} \frac{1}{h} \int_{1}^{1+2h} e^{\sqrt{x}} \sin\left(\frac{\pi x}{3}\right) dx$ equals:

Let $I(x) = \displaystyle\int \frac{dx}{(x-11)^{11/13}(x+15)^{15/13}}$. If $I(37) - I(24) = \dfrac{1}{4}\!\left(\dfrac{1}{b^{1/13}} - \dfrac{1}{c^{1/13}}\right)$, $b, c \in \mathbb{N}$, then $3(b+c)$ is equal to

If $\displaystyle\int \frac{2x^2+5x+9}{\sqrt{x^2+x+1}}\,dx = x\sqrt{x^2+x+1} + \alpha\sqrt{x^2+x+1} + \beta\log_e\!\left|x+\tfrac{1}{2}+\sqrt{x^2+x+1}\right| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ____.

If $\displaystyle\int e^x\!\left(\frac{x\sin^{-1}x}{\sqrt{1-x^2}} + \frac{\sin^{-1}x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2}\right)dx = g(x)+C$, where $C$ is the constant of integration, then $g\!\left(\dfrac{1}{2}\right)$ equals:

If the value of the integral $\int_{-1}^{1}\dfrac{\cos\alpha x}{1+3^x}\,dx$ is $\dfrac{2}{\pi}$, then a value of $\alpha$ is:

If $\displaystyle\int\frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}}\,dx=A\!\left(\dfrac{\alpha x-1}{\beta x+3}\right)^{\!B}+C$, where $C$ is the constant of integration, then the value of $\alpha+\beta+20AB$ is

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

If $\int \cosec^5 x\,dx = \alpha\cot x\cosec x\!\left(\cosec^2 x+\dfrac{3}{2}\right)+\beta\log_e\!\left|\tan\dfrac{x}{2}\right|+C$, where $\alpha,\beta\in\mathbb{R}$ and $C$ is the constant of integration, then the value of $8(\alpha+\beta)$ equals

Integer AnswerIf $\displaystyle\int x^2 e^{3x}\,dx = e^{ax}\left[bx^2 - cx + d\right]+c$, then $\dfrac{abc}{d}=$

Let $[\cdot]$ be the greatest integer function. If $\alpha=\displaystyle\int_0^{64}\left(x^{1/3}-\left[x^{1/3}\right]\right)dx$, then $\dfrac{1}{\pi}\displaystyle\int_0^{\alpha\pi}\left(\frac{\sin^2\theta}{\sin^6\theta+\cos^6\theta}\right)d\theta$ is equal to _____

Let $[\cdot]$ denote the greatest integer function. Then $\displaystyle\int_{-\pi/2}^{\pi/2}\left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right)dx$ is equal to:

The number of elements in the set $S=\left\{x:x\in[0,100]\text{ and }\displaystyle\int_0^x t^2\sin(x-t)\,dt=x^2\right\}$ is _____

$6\displaystyle\int_0^\pi|\sin 3x+\sin 2x+\sin x|\,dx$ is equal to _____

Let a differentiable function $f$ satisfy the equation $\displaystyle\int_0^{36}f\!\left(\frac{tx}{36}\right)dt=4\alpha f(x)$. If $y=f(x)$ is a standard parabola passing through the points $(2,1)$ and $(-4,\beta)$, then $\beta^\alpha$ is equal to _____

Evaluate the following limit: 8. $\lim_{n \to \infty} \frac{1}{n}\left(\sin\frac{\pi}{2n} + \sin\frac{2\pi}{2n} + \sin\frac{3\pi}{2n} + \cdots + \sin\frac{n\pi}{2n}\right)$

955. If $\displaystyle\int_{-39}^{59} \dfrac{\sin(2(\{x\} + \{-x\}))}{e^{-\{x\}}} \left(\dfrac{\tan x - \tan[x]}{1 + \tan x \tan[x]} + \sec^2\{x\}\right) dx = p \cdot e \cdot \sin^2 q$ where $p, q \in N$ and $e$ is Napier's constant, then find the value of $(p + q)$.[Note: $[k]$ and $\{k\}$ denotes greatest integer function less than or equal to $k$ and fractional part function of $k$ respectively.]

892. Let $f$ be a continuous and even function such that $\int_0^a f(x)\,dx = 10$. If $g(x)$ is a continuous positive function such that $g(x)g(-x) = 1$ and $\int_0^a g(x)\,dx = 5$, then find the value of $\displaystyle\int_{-a}^{a} \dfrac{f(x)}{1+g(x)}\,dx$.

911. If $f$ and $g$ are two functions such that $2f(1)=g(2)=4$ and $2f(9)=g(10)=20$ and $\int_0^2 (x^2 g(f(x^3+1))f'(x^3+1)-3x^2)\,dx = 0$, then find the value of $\int_4^{20} g^{-1}(x)\,dx$.

$\lim_{n \to \infty} \frac{1}{n} \sum_{r=n+1}^{2n} \log\left(1 + \frac{r}{n}\right)$ equals

Evaluate $\int e^{\tan^{-1} x} \left( \frac{1+x+x^2}{1+x^2} \right) dx$

If $24 \int_0^{\pi/12} \left(\left|\sin\left|4x - \frac{\pi}{12}\right|\right| + [2\sin x]\right) dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to _______.

Evaluate $$\int \frac{2 \sin 2x - \cos x}{6 - \cos^2 x - 4 \sin x} \, dx$$

Let $I_n = \int_0^{\pi/4} \tan^n x dx$ ($n > 1$ and is an integer). Show that $\frac{1}{2(n+1)} < I_n < \frac{1}{2(n-1)}$.

Let $I_n = \int_{0}^{1} 1 \cdot (1-x^4)^n dx, n \in N$ then prove that $\frac{I_n}{I_{n-1}} = \frac{4n}{4n+1}$

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

$\displaystyle\int \frac{x}{(x^2+1)(x^2+4)}\,dx$ equals

The derivative of $x^4 + x^{-5}$ is $-\left(4x^{-5} + 5x^{-6}\right)$. So, $$\int \frac{5x^3 + 4x^5}{\left(x^5 + x + 1\right)^2} dx =$$

If $f(x) = \lim_{n \to \infty} \left[2x + 4x^3 + ... + 2nx^{2n-1}\right]$ $(0 < x < 1)$ then $\int f(x)dx$ is equal to:

Evaluate $I = \displaystyle\int_{e^{\pi/6}}^{e^{\pi/2}} \frac{\sin(\ln(\sin(\ln x)))\cdot\cos(\ln x)}{x\sin(\ln x)}\,dx$. Find $\cos^{-1}(I+1)$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \frac{x^2 - 3x - 6}{x^2 + 2x + 4}$. Then which of the following statements is (are) TRUE ?

$$\int_1 \frac{(2x^3 + 3x^2 - 1)\,\sqrt{4x^6 + 12x^5 + 9x^4 - 6x^3 - 9x^2 + 2}}{x(x+1)}\,dx =$$

For non-negative integers $a$ and $b$, let $I(a,b)=\displaystyle\int_0^{\pi/2}\cos^a x\cos bx\,dx$. Match each entry in List-I to the correct entry in List-II. **List-I:** P) $I(0,5)$; Q) $I(1,4)$; R) $I(2,4)$; S) $I(3,2)$ **List-II:** 1) $\dfrac{1}{5}$; 2) $\dfrac{1}{5}I(2,1)$; 3) $\dfrac{1}{4}[I(1,3)-I(1,5)]$; 4) $\dfrac{1}{5}I(0,3)$; 5) $\dfrac{1}{3}I(2,1)$

If $\displaystyle\int_0^{4\pi}\ln\left|\frac{1}{3}\sin x+\sqrt{3}\cos x\right|\,dx = k\pi\ln 7$, then the value of $k$ is:

Definite Integration Questions (1340)