Definite Integration Questions - JEE

10.Column-IColumn-II(A) Let $f(x) = \int x^{\sin x} (1 + \cos x \ln x + \sin x) dx$ and $f\left(\frac{\pi}{2}\right) = \frac{\pi^2}{4}$ then the value of $f(\pi)$ is(P) rational(B) Let $g(x) = \int \frac{1 + 2 \cos x}{(\cos x + 2)^2} dx$ and $g(0) = 0$ then the value of $g\left(\frac{\pi}{2}\right)$ is(Q) irrational(C) Let $k(x) = \int \frac{(x^2 + 1) dx}{\sqrt[3]{x^3 + 3x + 6}}$ and $k(-1) = \frac{1}{\sqrt{2}}$ then the value of $k(-2)$ is(R) integral(D) 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 and $f(x)$ is positive), then $f(0) + g(0)$ is(S) prime

$\int \frac{a+b\cos x}{(b+a\cos x)^2} dx$ is equal to

Let f(x) = e√x sin(πx/3) dx and F(x) = ∫0x f(t) dt. Then the value of $L = \lim_{h \to 0} \frac{1}{h} \int_{1}^{1+2h} e^{\sqrt{x}} \sin\left(\frac{\pi x}{3}\right) dx$ is

$\int \frac{x+1}{x\sqrt{x+1}} dx$ is equal to

If the value of $\lim_{n \to \infty} \sum_{k=0}^{n} \dfrac{{}^n C_k}{n^k(k+3)}$ equals $L$. Then $[L]$ is equal to:[Note: Where $[k]$ denotes greatest integer function less than or equal to $k$.]

Let $f: [-1, 0] \to \mathbb{R}$ be a function differentiable within the domain and that $\displaystyle\int_{-1}^{0} (f(x))^2\, dx = 10$ and $f(-1) = 2$. The value of the integral $\displaystyle\int_{-1}^{0} x f'(x) f(x)\, dx$ is:

Let f be a real valued function satisfying $ f(x) + f(1-x) = \dfrac{9^x}{9^x+3} + \dfrac{9^{1-x}}{9^{1-x}+3} $. Find the value of $ \displaystyle\sum_{r=1}^{2008} f\!\left(\frac{r}{2009}\right) $.

$\int \frac{\sin 2x}{\sin^4 x + \cos^4 x} dx = a\cot^{-1}(b\tan 2x) + c$, then

Evaluate \[I = \int_0^{10\pi} |\sin x|\,dx\]

If $\int f(x)\,dx = g(x)$, then $\int f(x^{-1})\,dx$ is equal to

Evaluate: $I = \displaystyle\int_{-\pi}^{\pi} \dfrac{2x(1+\sin x)}{1+\cos^2 x}\, dx$

A strictly increasing continuous function $f(x)$ intersects with its inverse $f^{-1}(x)$ at $x = \alpha$ and $x = \beta$. If $\displaystyle\int_{\alpha}^{\beta}(f(x)+f^{-1}(x))\,dx = 13$ where $\alpha,\beta \in N$, then the value of $|\alpha\beta|$ equals:

Evaluate $\int \sqrt{\frac{3-x}{3+x}} \cdot \sin^{-1}\left(\frac{1}{\sqrt{6}} \sqrt{3-x}\right) dx$

The value of $\int_{-\pi}^{\pi} \frac{2x(1+\sin x)}{1+\cos^2 x} dx$ is:

Let $I_1 = \int_0^1 2^{x^2}\,dx$, $I_2 = \int_0^1 2^{x^3}\,dx$, $I_3 = \int_1^2 2^{x^2}\,dx$, $I_4 = \int_1^2 2^{x^3}\,dx$. Which of the following is correct?

The integral $\displaystyle\int \dfrac{dx}{(x+1)^{3/4}(x-2)^{5/4}}$ is equal to

Let $P_k$ be a point in xy-plane whose x coordinate is $1 + \dfrac{k}{n}$ $(k = 1, 2, 3, \ldots, n)$ on the curve $y = \ln x$. If A is (1, 0) then $\displaystyle\lim_{n\to\infty} \dfrac{1}{n} \sum_{k=1}^n (AP_k)^2$ equals

If $\int_{0}^{\pi} xf(\sin x) \, dx = A \int_{0}^{\pi/2} f(\sin x) \, dx$, then $A$ is

Let $f: [-1, 0] \to \mathbb{R}$ be a function differentiable within the domain and that $\displaystyle\int_{-1}^{0} (f(x))^2 \, dx = 10$ and $f(-1) = 2$. The value of the integral $\displaystyle\int_{-1}^{0} x f'(x) f(x) \, dx$ is:

The value of the integral $I = \int_0^1 x(1-x)^n\, dx$ is

We have \[I = \int\left\{\frac{(\log x - 1)}{1 + (\log x)^2}\right\}^2 dx\] Then $I$ equals:

Find the value of: \[ L = \lim_{n \to \infty} \sqrt[n]{\frac{n^{2n}}{(n^2+1^2)(n^2+2^2)\cdots(n^2+n^2)}} \]

The value of $\int_{-2}^{3} |1 - x^2| \, dx$ is

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

$A = 4\int_0^1 x\sqrt{1 - x^2}\, dx$. Find the value of $A$.

The value of the integral $\displaystyle\int_4^{10} \dfrac{[x^2]\,dx}{[x^2-28x+196]+[x^2]}$, where $[x]$ denotes the greatest integer less than or equal to $x$, is

Let \[ I = \int_{2}^{3} \frac{x^2\, dx}{2x^2 - 10x + 25} \] Find the value of $I$.

Let \[I = \int e^{\sin x}\left(\frac{x\cos^3 x - \sin x}{\cos^2 x}\right)dx\]If $I = e^{\sin x}\cdot f(x) + C$, then $f(x) = x$ and find the value of $\dfrac{f(7)}{2}$.

$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} \frac{r}{\sqrt{n^2 + r^2}}$ equals

If $\int \frac{\sqrt{1-x^2}}{x^4} dx = A(x)(\sqrt{1-x^2})^m + C$, for a suitable chosen integer $m$ and a function $A(x)$, where $C$ is a constant of integration, then $(A(x))^m$ equals:

The value of $ a_0 = \dfrac{1}{\pi} \int_{-\pi}^{\pi} x + x^2 \, dx $ is:

The value of $\int_0^{1/2} \dfrac{\ln(1+2x)}{1+4x^2}\,dx$ is:

If $\int_0^{\pi} x f(\sin x)\,dx = A\int_0^{\pi/2} f(\sin x)\,dx$, then the value of $A$ is:

If $f(x) = \displaystyle\int \dfrac{(3x^4 - 1)}{(x^4 + x + 1)^2}\, dx$ and $f(0) = 0$, then $f(-1)$ is equal to:

The value of $\int_0^1 \frac{(\sqrt{1-x^2})}{x^2} dx$ is equal to

$\int_{-\pi/2}^{-\pi/2} [(x+\pi)^3 + \cos^2(x+3\pi)]\, dx$ is equal to

If $A_n = \int_0^\pi \sin x dx$, $\forall n \in \mathbb{N}$, then $\sum_{n=1}^\infty A_n$ is equal to

The value of $\int_0^{\pi} \sec^2 x\, dx$ is

Let $f(x) = \dfrac{e^x}{1+e^x}$. If $I_1 = \int_{f(-a)}^{f(a)} x\,g\{x(1-x)\}\,dx$ and $I_2 = \int_{f(-a)}^{f(a)} g\{x(1-x)\}\,dx$, then find the value of $\dfrac{I_2}{I_1}$.

The value of the definite integral \[ \int_{1/3}^{1} \frac{\pi \cos\!\left(\dfrac{2\pi}{3}x\right) + \pi \cos\!\left(\dfrac{\pi}{3}x\right)}{\sin\!\left(\dfrac{\pi}{2}x\right)\sin\!\left(\dfrac{2\pi}{3}x\right) + 2\sin\!\left(\dfrac{\pi}{2}x\right)\sin\!\left(\dfrac{\pi}{3}x\right)} \, dx \] is equal to:

Let $f(x) = \int_0^x \ln(1+t^2)\, dt + x\ln(1+x^2)$. Then $f''(0)$ equals:

$\int \frac{dx}{\cos x - \sin x}$ is equal to

We have $F(x) + F\!\left(x + \dfrac{1}{2}\right) = 3$. Find the value of $\int_0^2 F(x)\, dx$.

The integral $\int_0^{[x]} (x - [x] + (\frac{1}{x}))dx$ is equal to (where $[x]$ is the greatest integer $\leq x$)

If $f(t)$ is an odd function, then prove that $\phi(x) = \int_0^x f(t)\,dt$ will be an even function.

The value of $\lim_{n \to \infty}\left(\ln\left(\sqrt[n]{\dfrac{4}{n^2}}\right) + \ln\left(\sqrt[n]{\dfrac{16}{n^2}}\right) + \ln\left(\sqrt[n]{\dfrac{36}{n^2}}\right) + \ldots + \ln\left(\sqrt[n]{\dfrac{4n^2}{n^2}}\right)\right)$ equals:

The value of $\displaystyle\int_0^{\pi/2} \log \tan x\,dx$ is ______.

If $f(x)$ is an even function, then the value of $\int_{-2}^{2} [x^2 f'(x) - x^3 f''(x) - \lambda] dx$, where $\lambda$ is a constant, is

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

If $\displaystyle\int_0^x f(t)\,dt = e^x - ae^{2x}\int_0^1 f(t)e^{-t}\,dt$, then $f(1) + 2f(2)$ is equal to:

Definite Integration Questions (1340)