Definite Integration Questions - JEE

Let f(x) = ∫−1x (2 − t2)dt. The real roots of the equation x2 + f′(x) = 0 are

Evaluate $I = \int \dfrac{1}{2e^{2x} + 3e^x + 1}\,dx$. The answer involves a logarithmic expression. Find the integer value associated with the result.

If $x^2 d(\tan^{-1} x) = x f(x) + c$, then $f(1)$ is equal to

Evaluate $\int e^x dx$

Evaluate: $\int \frac{x^3 + 3x^2 + x + 9}{(x^2 + 1)(x^2 + 3)} dx$

The integral \int \frac{2x dx}{(2x^4 + 3x^2 + 1)^4} is equal to (where C is a constant of integration)

Show that $\int_0^{\pi/2} \sin x \cdot \log \sin x \, dx = \log_e 2 - 1$.

If an anti-derivative of $f(x)$ is $e^x$ and that of $g(x)$ is $\cos x$, then $\int f(x)\cos x \, dx + \int g(x) e^x \, dx$ is equal to

Evaluate $I = \int \frac{(\sin 2x)^{1/3}}{(\sin^{2/3} x + \cos^{2/3} x)^2} d(\tan^{1/3} x)$

$\int \frac{x(\log x)^m}{(\log x)^m}\,dx$ is equal to

If $\int \sin^{-1}\left(\frac{x}{1+x}\right) dx = A(x) \tan^{-1}(\sqrt{x}) + B(x) + C$, where $C$ is a constant of integration, then the ordered pair $(A(x), B(x))$ can be

$\int \frac{dx}{(2x - 7)\sqrt{x^2 - 7x + 12}}$ is equal to

Prove the reduction formula: $$\int \sec^n x \, dx = \frac{1}{n-1} \sec^{n-2} x \tan x + \frac{n-2}{n-1} \int \sec^{n-2} x \, dx$$

Evaluate: $\int x \sin x \sec^3 x dx$

Let f(x) = \int x^2 \cos 2x (2x + 6\tan x - 2x\tan^2 x) dx and f(x) passes through the point (\pi, 0).If f: \mathbb{R} - \{(2n+1)\frac{\pi}{2}\} \to \mathbb{R} then f(x) be a:

Evaluate $\int e^x \sec x(1 + \tan x) dx$

Let $ I = \int \dfrac{\left(\sin^{3/2}\theta + \cos^{3/2}\theta\right) d\theta}{\sqrt{\sin^3\theta \cos^3\theta \sin(\theta+\alpha)}} $. Then $I$ equals:

Find $\int \frac{2^x + 3^{2x+1} + 5^{3x-2}}{7^x} dx$

Evaluate: \ 10\int \cos\left(\frac{x+1}{\sqrt{x^2+2x+5}}\right) dx

[JEE Main 2022] $\displaystyle\int\frac{\cos^2 x}{(\cos x+\sin x)^3}\,dx$ equals (where $C$ is a constant)

$\int e^{6\log x - e^5\log x}}{e^{4\log x - e^3\log x}} dx$ is equal to

Assertion (A): When $f(x) = \frac{x^2 + 1}{2}$, $\int \frac{dx}{x} = 2\ln|x| + c$Reason (R): $\int (h(x))^n h'(x) dx = \frac{(h(x))^{n+1}}{n+1} + C$

If $f(x) = f(x) + xf'(x)$ then $\int g(x)\,dx$ is equal to:

Let $n \geq 2$ be a natural number and \(0

Evaluate $\int \frac{1 + x}{\sqrt{1+x^4}} dx$

The value of $\displaystyle \int \frac{secx(2+secx)}{(1+2secx)^2} dx$ is equal to

The value of $\displaystyle \int \frac{dx}{(1+x) \sqrt{x} \sqrt{1-x}}$ is equal to

Given \[ I = \int \frac{\tan x}{1 + \tan x + \tan^2 x}\, dx = Kx + A\tan^{-1}\left(\frac{2\tan x + 1}{\sqrt{3}}\right) + C \]Find the values of $K$ and $A$.

Evaluate $\int \frac{f(x)g'(x) - f'(x)g(x)}{f(x)g(x)} \left[\log(g(x)) - \log(f(x))\right]\, dx$

[JEE Main 2020] $\displaystyle\int\frac{\sin x+\cos x}{\sqrt{1-\sin 2x}}\,dx$ for \(0

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

∫ (2+√x)dx/x²(√(x+1+√x)) is equal to:

$\int \frac{\sin 2x}{\cos 7x \sqrt{1 + 2\cos 5x \cos 8x}} dx$ is equal to

If $\int \frac{dx}{(\sin x + 4)(\sin x - 1)} = A \tan^{-1}\left(\frac{\tan x/2 - 1}{2}\right) + B \tan^{-1} f(x) + C$, then find the value of A.

67. $\int \frac{1 + \sin x}{\cos x} \, dx$ equals

If the area bounded between x-axis and the graph of y = 6x – 3x² between the ordinates x = 1 and x = a is 19 square units then 'a' can take the value

Evaluate: $\int \frac{dx}{2 + \sin^2 x}$

If $I(x)=\displaystyle\int e^{\sin^2x}(\cos x\sin 2x-\sin x)\,dx$ and $I(0)=1$, then $I\!\left(\dfrac{\pi}{3}\right)$ is equal to

Find $\int \frac{2x^2 + 3x^5}{1+x^6} dx$

[JEE Main 2023] If $\displaystyle\int x^5 e^{-x^2}\,dx = e^{-x^2}\cdot f(x)+C$ and $f(0)=-120$, what is the integer value of $f(1)$?

Assertion (A): If the primitive of $f(x) = \sin x + 2x - 4$ has the value 3 for $x = 1$, then there are exactly two values of $x$ for which the primitive of $f(x)$ vanishes.Reason (R): $\cos x$ has period $2\pi$.

If \int 2 2x +5x+9 dx = x\sqrtx 2 + x + 1 + \alpha\sqrtx 2 + x + 1+ \beta log ∣ e∣ x + 1 2 + \sqrtx 2 + x + 1∣ ∣ + C , where C is the \sqrtx2 +x+1 constant of integration, then \alpha + 2\beta is equal to _______.

Let $f(x)$ be a function satisfying $f'(x) = f(x)$ and $f(0) = 2$. Then $\int \frac{f(x)}{3 + 4f(x)} dx$ is

Evaluate $\int \frac{dx}{2x^2+x-1}$

If $I = \int_{-1}^{1} \frac{\cos^{-1}\left(\frac{x^4}{4}\right)}{1+x^2} dx = k\int_{0}^{1} \frac{\cos^{-1}\left(\frac{x^4}{4}\right)}{1+x^2} dx$, then find $k$.

If A(n) represents the area bounded by the curve y = n ln x, where n ∈ ℕ and n > 1, the x-axis and the lines x = 1 and x = e, then the value of A(n) + nA(n–1) is equal to:

$\displaystyle\int\frac{dx}{(x-\alpha)(x-\beta)}$ equals $(\alpha\neq\beta)$

Evaluate $\int \frac{1 - x^2}{x(1 - 2x)} dx$

Evaluate $\int \frac{dx}{\sqrt{(x-a)(b-x)}} (b > a)$

If $f(x) = \lim_{n \to \infty} \frac{x^n - x^{-n}}{x^n + x^{-n}}$, $0 < x < 1$, $n \in \mathbb{N}$ then $\int (\sin^{-1} x) f'(x) dx$ is equal to:

Definite Integration Questions (1340)