Definite Integration Questions - JEE

If $m = \int_0^{\pi/2} \frac{|\sin x|}{x + 1} dx$ and $n = \int_0^{\pi/2} \frac{|\sin x|}{x + 1} dx$ (where $[\cdot]$ = G.I.F.), then

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

If I_n = ∫(sin x)^n dx n ∈ N, then 5I_4 - 6I_6 is equal to -

The value of ∫010 sgn(x − [x]) dx, where [·] denotes the greatest integer function, is equal to

If $x$ satisfies the equation $\int_x^{2x} \frac{1 + t^2 \sin^2 t}{t^2} dt = ...$, then

If ∫₋π/₄^(π/4) e^(e^(sin x))(sin x + cos x)dx / (e^x + e^(−x)) = k∫₋π/₂^(π/2) sec x dx, then the value of k is

Primitive of $\frac{3x^4 - 1}{(x^4 + x + 1)^2}$ w.r.t. $x$ is -

If f(x) = x + 1 + \frac{2}{2!} + \frac{3}{3!} + \ldots, \int_0^2 (f(t))^2 \, dt = 6, and \int_0^2 (f(t)) \, dt = 2, then f(9) is equal to

Calculate $A_1 + A_2 = \int_0^{\pi/2} \cos x \, dx$.

Let f(x) = x / (1 + x^n)^(1/n) for n >= 2 and g(x) = (f o f o ... o f)(x) (f occurs n times). Then ∫ x^(n-2) g(x) dx equals.

∫ (x^2 - 1) / (x^3 * sqrt(2x^4 - 2x^2 + 1)) dx is equal to -

Evaluate $I = \displaystyle\int_{0}^{\pi/2} \sin^2 t \ln(\sin t)\,dt$. If $I$ can be expressed as a rational multiple of $\pi$, find the value of $16I/\pi$ (the answer is an integer).

The value of $\int_0^1 (1 + e^{-x^2}) dx$ is

727. Let $f(x)$ be monotonically strictly increasing function in $[3, 5]$ such that $\int_3^5 f^2(x)\,dx = 9$; $f(1) = 3$; $f(4) = 5$. Find the value of $2\int_1^4 x(5 - f^{-1}(x))\,dx$.

\sum_{r=1}^{10} \int_{0}^{1} f(r-1+x)\,dx is equal to

Let f(x) and g(x) be continuous, positive functions such that $f(-x) = g(x) - 1$, $f(x) = \dfrac{g(x)}{g(-x)}$ and $\displaystyle\int_{-20}^{20} f(x)\,dx = 2020$, then the value of $\displaystyle\int_{-20}^{20} \dfrac{f(x)}{g(x)}\,dx$ is:

Given \[\int_{-\pi/2}^{\pi/2} \frac{dx}{[x] + [\sin x] + 4}\]where $[\cdot]$ denotes the greatest integer function. The value of the integral is:

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

The graph of f(x) = x2 + ax + b intersects the x-axis at 2 distinct points A, B and y-axis at C. The centroid of $\triangle$ABC lie on the line y = x. If I = $\int_\limits{0}^{6}$ f(x) dx, then the value of I cannot be :

Evaluate $\int_{0}^{16\pi/3} |\sin x| dx$

Integral of $\sqrt{1+2\cot x(\cot x+\csc x)}$ w.r.t. $x$ is

The integral $\displaystyle\int_{\pi/4}^{3\pi/4} \dfrac{dx}{1+\cos x}$ is equal to

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

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

On the interval , find the least value of the function .

We have $ I = \int_0^{2\pi} [\sin 2x(1+\cos 3x)]\,dx $. Find the value of $I$.

The integral ∫ (x^2 - 1) / (x^3 * sqrt(2x^4 - 2x^2 + 1)) dx is equal to -

The integral \int_{\pi/3}^{\pi/6} 3\tan^7 3x \cdot \sin^2 3x (2\sec^2 x \cdot \sin^2 3x + 3\tan x \cdot \sin^6 x) dx is equal to

The value of $\lim_{n \to \infty} \sum_{k=1}^{n} \cos\left(\frac{n n-k}{n}\right) \cdot \frac{4^k}{n^2}$ equals:

$\int \frac{5x^4 + 4x^5}{(x^5 + x + 1)^2} dx$ equals

If $\int_0^2 (3x^2 - 3x + 1)\cos(x^3 - 3x^2 + 4x - 2) dx = a \sin(b)$, where $a$ and $b$ are positive integers, find the value of $(a + b)$.

The value of the definite integral $\displaystyle\int_0^{\pi/4} \dfrac{\sin^3 x \cos^3 x}{(\sin^4 x + \cos^4 x)^2}\, dx$ is equal to:

The value of definite integral \[\int_0^{\infty} \frac{dx}{(1+x^9)(1+x^2)}\]

If $I_1 = \displaystyle\int_0^1 e^{-x}\cos^2 x\, dx$, $I_2 = \displaystyle\int_0^1 e^{-x^2}\cos^2 x\, dx$ and $I_3 = \displaystyle\int_0^1 e^{-x^3}\, dx$ then

If $\int_{1}^{xy} f(t)\,dt = y\int_{1}^{x} f(t)\,dt + x\int_{1}^{y} f(t)\,dt$ for all $x, y \in \mathbb{R} - \{0\}$ and $f(1) = 1$, and $g(x) = -\!\left(x^2 + \dfrac{1}{x^2}\right)$, find the value of $I = \displaystyle\int_{0}^{\infty} e^{-\left(x^2+\frac{1}{x^2}\right)}dx$ (multiplied by an appropriate factor as given in the solution, answer = 10).

Evaluate \int_0^{np+w} |\sin x| \, dx, where n \in \mathbb{N} and 0 \leq w

The integral ∫ (x^2 - 1) / (x^3 * sqrt(2x^4 - 2x^2 + 1)) dx is equal to -

\int_{-1}^{1} f'(1+x^2) x^2 e^{-\cos \pi t} dx is equal to:

$100\int_0^1 \{x\} dx$, where $\{x\}$ denotes the fractional part of $x$

Integral of $\sqrt{1+2\cot x(\cot x+\csc x)}$ w.r.t. $x$ is

The value of the definite integral \[\int_0^{\pi/2} \frac{1 + \sin^3 x}{1 + 2\sin x} dx\]

The value of ∫π/20 \frac{\sin x}{\sin x + \cos x} dx is

If $A = \int_0^\pi \sin x \, dx$, then $\int_0^\pi \frac{\cos^2 x}{x} dx$ is equal to:

The value of the definite integral \[\int_0^{\pi/2} \frac{dx}{\tan x + \cot x + \csc x + \sec x}\]

If f(x) = ax^3 + bx^2 + cx + d, where a, b, c and d are real numbers and 3b^2 > 9ac, is an increasing cubic function and g(x) = af'(x) + bf''(x) + c^2, then \int_a^x g(t) dt is

Let f be integrable over [0, a] for any real values of a. If I1 = ∫0π/2 f(sin2θ)(sin2θ + cos2θ) dθ and I2 = ∫0π/2 sin 2θ f(sin2θ + cos2θ) dθ, then

If $\displaystyle\int_1^2 \dfrac{dx}{(x^2-2x+4)^{3/2}} = \dfrac{k}{k+5}$, then $k$ is equal to

The value of $\lim_{n \to \infty} \left(\frac{1}{2n+1} + \frac{1}{2n+2} + \frac{1}{2n+3} + \cdots + \frac{1}{3n}\right)$ is equal to

By Leibnitz Theorem, if $\dfrac{d}{dx}\int_{0}^{x^3} k(t)\,dt = \dfrac{d}{dx}\left(x^{1+x^2}\right)$, find the value of $3k(1)$.

If $\int_a^b |\sin x| dx = 8$ and $\int_{a+b}^b |\cos x| dx = 9$, then the value of $\int_a^b \frac{x \sin x}{2x} dx$ is:

Definite Integration Questions (1340)