Definite Integration Questions (1340)

Evaluate the following: 25. \(\int_0^{\pi/2} \cos 2x \cdot \log(1 + \tan x) \, dx\)

Find $I = \int_0^1 \frac{\sqrt{1-x}}{\sqrt{x} + \sqrt{1-x}} dx$ (rationalising the denominator)

The value of $\int_{-\pi}^{\pi}\dfrac{2y(1+\sin y)}{1+\cos^2 y}\,dy$ is:

Let $f:[0,2]\to\mathbb{R}$ be defined as $f(x)=\begin{cases}e^{\min\{x^2,x-[x]\}}, & x\in[0,1)\\e^{[x-\log_e x]}, & x\in[1,2]\end{cases}$. Then $\displaystyle\int_0^2 x f(x)\,dx$ is

If $24\displaystyle\int_0^{\pi/6}\left(\sin\!\left|4x-\frac{\pi}{12}\right|+\left[2\sin x\right]\right)dx = 2\pi+\alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ____.

If $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $\displaystyle\int_0^{\pi/2}f(\sin 2x)\sin x\,dx+\alpha\int_0^{\pi/4}f(\cos 2x)\cos x\,dx=0$, then the value of $\alpha$ is

280. 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 $I_1 = \int_0^e x^2dx$ and $I_2 = \int_0^e 2x e^x dx$, then the value of $I_1 + I_2$ is equal to

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x) = \displaystyle\int_0^x tf(t)\,dt$. If $g(x^3) = x^6+x^7$, then value of $\displaystyle\sum_{r=1}^{15}f(r^3)$ is:

Let $f(x) = \displaystyle\int_0^{x^2}\frac{t^2-8t+15}{e^t}\,dt$, $x\in\mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are:

Let $[t]$ denote the greatest integer function. If $\displaystyle\int_0^{2.4}[x^2]\,dx=\alpha+\beta\sqrt{2}+\gamma\sqrt{3}+\delta\sqrt{5}$, then $\alpha+\beta+\gamma+\delta$ is equal to

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

Let f : (0, \infty) \to R be a twice differentiable function. If for some a̸ = 0, \int 1 f (\lambdax)d\lambda = 0 ​ ​ af (x), f (1) = 1 and f (16) = 18 , then 16 - f ′ ( 16 1 ) is equal to _______. ​ ​

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

The integral $80\displaystyle\int_0^{\pi/4}\left(\frac{\sin\theta+\cos\theta}{9+16\sin 2\theta}\right)d\theta$ is equal to:

\pi sin 2 x 21 If I = \int , then \int equals : 2 x sin x cos x dx dx 0 3 3 0 4 4 sin x+cos x sin 2 x+cos 2 x 2

The value of the integral $\displaystyle\int_{\pi/24}^{5\pi/24}\frac{dx}{1+\sqrt[6]{\tan 2x}}$ is:

The value of $\displaystyle\int_{e^2}^{e^4}\frac{1}{x}\left(\frac{e^{(\log_e x)^2+1}-1}{e^{(\log_e x)^2+1}-1+e^{(6-\log_e x)^2+1}-1}\right)dx$ is

If \(f\) is continuous and differentiable on \([0,1]\) with \(f(0)=0\) and \(f(1)=1\), then the minimum value of \(\displaystyle\int_0^1 (f'(x))^2\,dx\) is equal to

If the value of the integral $I = \int_{-\pi}^{\pi} \max(\sin x, \tan x)dx$ is equal to $\ln k$, then the value of $k^2$ is equal to

Evaluate \(\displaystyle\int_0^{\pi}\frac{x}{1+\sin x}\,dx\) [JEE Main 2017]

If $\displaystyle\int_0^1\dfrac{1}{(5+2x-2x^2)(1+e^{2-4x})}\,dx=\dfrac{1}{\alpha}\log_e\!\left(\alpha+\dfrac{1}{\beta}\right)$, $\alpha,\beta>0$, then $\alpha^4-\beta^4$ is equal to

Let for f (x) = 7 tan x + 7 tan x - 3 tan x - 3 tan x, I = \int 8 6 4 2 1 0 \pi/4 f (x)dx and I = \int 2 \pi/4 0 xf (x)dx . Then 7I1 + 12I2 is equal to :

For $m,n>0$, let $\alpha(m,n)=\displaystyle\int_0^2 t^m(1+3t)^n\,dt$. If $11\alpha(10,6)+18\alpha(11,5)=p(14)^6$, then $p$ is equal to

Let $\beta(m,n)=\int_0^1 x^{m-1}(1-x)^{n-1}\,dx$, $m,n>0$. If $\int_0^1(1-x^{10})^{20}\,dx=a\cdot\beta(b,c)$, then $100(a+b+c)$ equals:

Let $f(x)=\dfrac{x}{(1+x^n)^{1/n}}$, $x\in\mathbb{R}\setminus\{-1\}$, $n\in\mathbb{N}$, $n>2$. If $f_n(x)=(f\circ f\circ\cdots$ up to $n$ times$)(x)$, then $\displaystyle\lim_{n\to\infty}\int_0^1 x^{n-2}(f_n(x))\,dx$ is equal to

Evaluate \(\displaystyle\int_1^e\frac{dx}{x(1+\ln x)^2}\) [JEE Main 2019]

Evaluate \(\displaystyle\int_0^1\frac{dx}{1+x^2}\)

If $\int_0^{\pi/2}\dfrac{\sin^2 x}{1+\sin x\cos x}\,dx=\dfrac{1}{a}\log_e\left(\dfrac{a}{3}\right)+\dfrac{\pi}{b\sqrt{3}}$, where $a,b\in\mathbb{N}$, then $a+b$ is equal to ________.

Let for some function $y = f(x)$, $\displaystyle\int_0^x tf(t)\,dt = x^2 f(x)$, $x>0$ and $f(2) = 3$. Then $f(6)$ is equal to

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x\in\mathbb{R}$, $\displaystyle\int_0^2 xF'(x)\,dx = 6$ and $\displaystyle\int_0^2 x^2 F''(x)\,dx = 40$, then $F'(2)+\displaystyle\int_0^2 F(x)\,dx$ is equal to:

Evaluate \(\displaystyle\int_{-2}^2|x+1|\,dx\)

If $\displaystyle\int_{-0.15}^{0.15}|100x^2-1|\,dx=\dfrac{k}{3000}$, then $k$ is equal to _____.

Let $f:(0,\infty)\to\mathbb{R}$ be a twice differentiable function. If for some $a \neq 0$, $\displaystyle\int_0^1 f(\lambda x)\,d\lambda = af(x)$, $f(1) = 1$ and $f(16) = \dfrac{1}{8}$, then $16 - f'\!\left(\dfrac{1}{16}\right)$ is equal to ____.

Let f (x) = \int 0 t t (t 2 - 9t + 20) dt, 1 \le x \le 5 . If the range of f is [\alpha, \beta], then 4(\alpha + \beta) equals :

Evaluate $$\int \frac{e^{2x}(1+2x)}{\sin(xe^{2x})} dx$$

Let $f$ be a continuous function satisfying $\displaystyle\int_0^{t^2}(f(x)+x^2)\,dx=\dfrac{4t^3}{3}$, $\forall t>0$. Then $f(\pi^2)$ is equal to

Question 25: Let \(I = \int \tan^6 x\, dx - \int \tan^4 x(\sec^2 x - 1)\, dx\). If \(I = \frac{\tan^5 x}{5} - \frac{\tan^3 x}{3} + A\tan x - x + D\) where \(A, B, C\) are constants with \(A = -\frac{1}{3}, B = 1, C = -1\), then \(A + B + C\) equals:

If $f(x)$ satisfies the relation $f(x)=e^x+\displaystyle\int_0^1(y+xe^y)f(y)\,dy$, then $e+f(0)$ is equal to _____

Let \(f(x)=\int_{0}^{x} g(t) d t\), where g is non-zero even function. If f (x + 5) = g (x), then \(\int_{0}^{x} f(t) d t\) equals

Evaluate \(\displaystyle\int_0^1\sqrt{1-x^2}\,dx\)

If $\displaystyle\int_0^1 4\cot^{-1}(1-2x+4x^2)\,dx = a\tan^{-1}(2)-\log_e(5)$, where $a,b\in\mathbf{N}$, then $(2a+b)$ is equal to _____

Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=\pi^2$, $\forall x\in\mathbb{R}$. Then $\displaystyle\int_0^\pi f(x)\sin x\,dx$ is equal to

Let f be a real valued continuous function defined on the positive real axis such that g(x) = \int . If x tf (t)dt 0 , then value of \sum is : 15 3 6 7 3 g (x ) = x + x f (r ) r=1

Let $f$ be a polynomial function such that $f(x^2+1)=x^4+5x^2+2$, for all $x\in\mathbb{R}$. Then $\displaystyle\int_0^3 f(x)\,dx$ is equal to

The value of $\displaystyle\int_{-\pi/6}^{\pi/6}\left(\frac{\pi+4x^{11}}{1-\sin\!(|x|+\pi/6)}\right)dx$ is equal to:

The value of ∫₋₁¹ [x[1 + sin πx] + 1] dx is ([·] denotes the greatest integer)

26. If the value of definite integral \(\displaystyle\int_{\pi/4}^{\pi/3} e^x \left(\dfrac{2 + \sin 2x}{1 + \cos 2x}\right) dx\) is expressed as \(e^{\frac{\pi}{a}}\left(be^{\frac{\pi}{c}} - 1\right)\), then the value of \(\dfrac{b^2 c}{a}\) is:

If $f(x)$ is integrable function such that $|f(x) - f(y)| \le |x^2 - y^2|, \forall \ x, y \in [a, b]$ then prove that $\left| \int_a^b \frac{f(x) - f(a)}{x + a} dx \right| \le \frac{(a - b)^2}{2}$.

Suppose that \(f:[0,1]\to\mathbb{R}\) has a continuous derivative and that \(\int_0^1 f(x)\,dx = 0\). Then for every \(\alpha \in (0,1)\), find the minimum value of \(\dfrac{\left|\int_0^\alpha f(x)\,dx\right|}{\max_{0\leq x\leq 1}|f'(x)|}\).