Definite Integration Questions - JEE

If $I_2 = \displaystyle\int_0^1 \left(\frac{x}{5+x}\right)^{7/2}\left(\frac{1-x}{5+x}\right)^{9/2}\frac{dx}{(5+x)^2}$ and $I_2 = \dfrac{1}{a \cdot 5^{9/2} \times 6^{7/2}} I_1$, find the value of $a$.

Given $f(x) = x^3 + px^2 + qx$ with conditions: $\int_1^3 g(x)\,dx + \int_3^1 g^{-1}(x)\,dx = 0$, $f(x)|_{\max} = f(\pi) = \frac{5+3}{3-1} = 4$, and $f'(x) = 3x^2 + 2px + q$ with $\frac{q}{3} = 3 \Rightarrow q = p$ and $\frac{-2p}{3} = 4 \Rightarrow p = -6$. Find $p + q$.

10. $\int \frac{\ln \left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} d x$ equals -

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

If $F(x) = \int_{x^2}^{x^3} \frac{1}{\ln t} dt$ then find $F'(e)$.

Evaluate $\int x \cos 2x \, dx$

Evaluate $\displaystyle\int_0^3[x^2]\,dx$ where $[\cdot]$ is GIF. [JEE Main 2017]

Let I(x) = ∫√(x+7)/x dx and I(9) = 12 + 7 loge 7. If I(1) = α + 7 loge (1+2√2), then α^4 is equal to________.

∫ 3x² + 1 / (x² - 1)³ dx equals (where K is constant of integration)

Evaluate $\displaystyle\int_0^{\pi/2} \sin^5 x\cos^4 x\,dx$

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 :

Evaluate $\displaystyle\int_1^e \frac{\ln x}{x}\,dx$

Evaluate $\displaystyle\int_0^{\pi/2}x\cos x\,dx$ [JEE Main 2018]

Evaluate $\int_{0}^{\pi} \frac{x dx}{1 + \cos^2 x}$

Let f(x) = If f(x) is continuous at x = 2, then the value of k is equal to:

If f(x) = x3 + 3x + 4 then the value of $\int_\limits{-1}^{1}$ f(x) dx + $\int_\limits{0}^{4}$ f-1 (x) dx equals:

The value of integral $\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{x}{1+\sin x} d x$ is

Evaluate: $I = \int \left(\frac{\ln x - 1}{(\ln x)^2 + 1}\right)^2 dx$

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

3 points O (0, 0), P(a, a2), Q (-b, b2) (a > 0, b > 0) are on the parabola y = x2. Let S1 be the area bounded by the line PQ and the parabola and let S2 be the area of the triangle OPQ, the minimum value of $\frac{S_{1}}{S_{2}}$ is :

$I_n = \int_0^{\pi/4} \tan^n x\, dx$, then $\lim_{n \to \infty} n(I_n + I_{n+2})$ equals

Evaluate the integral: \[I = \int \frac{\sin^2 x \cos^2 x}{[(\sin^2 x + \cos^2 x)(\sin^3 x + \cos^3 x)]^2} dx\]

∫₋₁² (1 + x)/(x² + 2x + 1) dx is equal to

If $\int_{0}^{1} \frac{1}{\left(5+2 x-2 x^{2}\right)\left(1+e^{(2-4 x)}\right)} d x=\frac{1}{\alpha} \log _{e}\left(\frac{\alpha+1}{\beta}\right)$, $\alpha, \beta \gt 0$, then $\alpha^{4}-\beta^{4}$ is equal to

∫ \frac{1-x^7}{x(1+x^7)} dx equals -

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

If $f(y) = e^y$, $g(y) = y;\, y > 0$ and $F(t) = \int_0^t f(t-y)g(y)\, dy$, then

We have $\frac{d}{dx}F(x) = \left(\frac{e^{\sin x}}{x}\right),\ x > 0$. If $I = \int_1^4 \frac{3}{x} e^{\sin x^3} dx$, then $I$ equals:

The value of $\displaystyle\int_{-\pi/2}^{\pi/2} \frac{dx}{[x]+[\sin x]+4}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is:

Let \[ I = \int_{0}^{2\pi} \frac{x\sin^{2n} x}{\sin^{2n} x + \cos^{2n} x}\, dx \] Find the value of $I$ (nearest integer).

Consider $I = \displaystyle\int_{-a}^{a} \frac{f(x)}{1+e^{2x+1}}\, dx$, where $f(x)$ is an even function. Evaluate: \[\int_{-4}^{4} \frac{x^2}{(x^2+16)(1+e^{x^3})}\, dx\]

The value of $\int \frac{1-\ln x}{x^2} dx$ is

Let $\dfrac{d}{dx}F(x) = \left(\dfrac{e^{\sin x}}{x}\right)$, $x > 0$. If $\displaystyle\int_1^4 \dfrac{3}{x} e^{\sin x^3} dx = F(k) - F(1)$, then one of the possible value of $k$ is

If $\int \sqrt{\sec 2x - 1} dx = \alpha \log_e \left| \cos 2x + \beta + \sqrt{\cos 2x \left( 1 + \frac{\cos 2x}{\beta} \right)} \right| + \text{constant}$, then $\beta - \alpha$ is equal to ____.

Evaluate $\displaystyle\int_0^1 x^4(1-x)^3\,dx$

If $I = \int_0^{\pi} \frac{\sin x(1+\sin x)e^{\sin x - \cos x}}{e^{-\cos x}+1}\,dx$and the value of $100\left(1+\frac{1}{4}\right)$ is computed, find the numerical value of $100\!\left(1+\frac{1}{4}\right)$.

The value of $\lim_{n \to \infty} \sum_{k=0}^{n} \dfrac{{}^nC_k}{n^k} \int_{0}^{1} x^{k+2}\, dx$ is

Evaluate \int_{50}^{100} \frac{\ln x}{\ln x + \ln(150-x)} dx

If $f(x) = \lim_{n \to \infty} \frac{\tan(1/n)\log(1/n)}{n}$, and $\int \frac{f(x)}{\sqrt{\sin^{11} x \cos x}} dx = g(x) + C$ (C being the constant of integration). Then:

$$\int \frac{x^4 - 2}{x^2\sqrt{x^4 + x^2 + 2}} dx =$$

Evaluate $\int \left(\frac{x+2}{x+4}\right) e^x dx$

A function $f(x)$ continuous on $\mathbb{R}$ and periodic with $2\pi$ satisfies $f(x) + (\sin x) f(x + \pi) = \sin^2 x$ then,

The integral $\int x\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)dx\;(x>0)$ is equal to

The value of \lim_{n \to \infty} \sum_{k=0}^n \int_0^{k+2/n} \frac{x^k}{k!} dx is:

If $\int \dfrac{\cos^4 x\,dx}{(\sin^3 x)(\sin^5 x + \cos^5 x)^{3/5}}$, and the result involves constants $A$ and $B$ such that $A = 5$ and $B = \dfrac{2}{5}$, find $AB$.

If $f'(x) = 5^{x} \cdot 5^{f(x)}$, then $k$ is:

If \[ f(x) = x + \frac{2}{3}x^3 + \frac{2}{3}\cdot\frac{4}{5}x^5 + \frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}x^7 + \cdots \infty \] then the area $A = \int_{1/2}^{\sqrt{3}/2} \frac{\sin^{-1}x}{\sqrt{1-x^2}}\,dx = \frac{\pi^2}{24}$. If $A = \frac{\pi^2}{a+b}$ where $a+b$ equals:

If $p, q, r, s$ are in arithmetic progression and $f(x) = \begin{vmatrix} p + \sin x & q + \sin x & p - r + \sin x \\ q + \sin x & r + \sin x & -1 + \sin x \\ r + \sin x & s + \sin x & s - q + \sin x \end{vmatrix}$ such that $\int_0^2 f(x) dx = -4$, then the common difference of the progression is:

Let $'x'$ be a real valued differentiable function satisfying $f\left(\frac{x}{y}\right) = f(x) - f(y)$ and $\lim_{x \to 0} \frac{f(1+x)}{x} = 3$. If the area bounded by the curve $y = f(x)$, the Y-axis and the line $y = 3$, where $x,y \in \mathbb{R}$ is $K$, then $K$ is____.

Let $h(x) = (fog)(x) + K$ where $K$ is any constant. If $\frac{d}{dx}(h(x)) = -\frac{\sin x}{\cos^2(\cos x)}$ if $f(x) = \int_{f(x)}^{g(x)} \frac{f(t)}{g(t)} dt$, where $f$ and $g$ are trigonometric functions then the value of $j(0)$ is equal to $(\cos(1) = .54)$ ____.

Definite Integration Questions (1340)