Indefinite Integration Questions - JEE

Evaluate $\int (\sin x + x \cos x) dx$

Given $I = \displaystyle\int \frac{2x^3 - 1}{x^4 + x}\,dx$. Then $I$ equals:

Anti-derivative of $$\frac{x - 1}{(x + 1)\sqrt{x^3 + x^2 + x}}$$ is:

If $\int f(x)\,dx = \Psi(x)$, then $\int x^5 f(x^3)\,dx$ equals:

Given\[ I = \int \frac{\sin\left(\dfrac{5x}{2}\right)}{\sin\left(\dfrac{x}{2}\right)}\, dx \]Evaluate $I$.

Evaluate the integral: \[I = \int \frac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3} dx\]

For $x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$, if $y(x)=\displaystyle\int\dfrac{\csc x+\sin x}{\csc x\sec x+\tan x\sin^2x}\,dx$ and $\lim_{x\to(\pi/2)^-}y(x)=0$, then $y\left(\dfrac{\pi}{4}\right)$ is equal to

If $\int \frac{1 - x^7}{x(1 + x^7)} dx = P \log|x| + Q \log|x^7 + 1| + C$, then:

If $I = \int \frac{dx}{(x-1)^4(x+2)^3} = k\sqrt[4]{x+2} + C$, then $k$ is equal to:

Evaluate: $I = \int \frac{\sin^8 x - \cos^8 x}{1 - 2\sin^2 x \cos^2 x} dx$

Evaluate: $\int \frac{\sin 2x}{(3 + 4\cos x)^3} dx$

If \int \frac{\log(x + \sqrt{1 + x^2})}{1 + x}dx = g \circ f(x) + \text{constant}, then (JEE Main 2016)

Find $\int \frac{3x+1}{(x+1)^3} dx$

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

$\int \frac{\tan x}{\sin x \cos x} dx$ is equal to [IIT - 1998]

$\int \frac{x+1}{(x^2+1)^{3/2}} dx$ equals

Find $\int \frac{dx}{(x-p)\sqrt{(x-p)(x-q)}}$

The value of $\displaystyle \int \frac{dx}{\sqrt{a^2-x^2}}$ is equal to

$\int \frac{\sec x}{(\sec x + \tan x)^{9/2}} dx$ equals (for some arbitrary constant K) [IIT - 2012]

Evaluate \int x^3 \sqrt[4]{1 + x^{1/3}} \, dx

If $f(x) = \sqrt{\frac{x + 2}{2x + 3}}$, then evaluate $\int \frac{f(x)}{x^{1/2}} dx$

$\int \frac{\sin 2x \cos 2x}{(\sin^5 x + \cos^3 x \sin^2 x + \sin^3 x \cos^2 x + \cos^5 x)^2} dx$ is equal to (JEE Main 2018)

$\int \frac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}\,dx$ is equal to

$\int x^m x^n - 1 + \frac{1}{x^n} - \frac{mx}{n} + \frac{nx}{m} - \frac{1}{2} - \frac{n}{2m} + \frac{m}{n} dx$ is equal to

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

If $\int f(x)\,dx = f(x) + C$, then $\int [f(x)]^2\,dx$ is

The value of $\int \frac{f(x)\phi'(x) - \phi(x)f'(x)}{[f(x)\phi(x)-1]^2 - f(x)\phi(x) - 1} dx$ is

Evaluate $\int \frac{\ln x}{x(1 + \ln x)} dx$

The value of $\displaystyle \int \frac{e^x + x^4 + 2}{(1+x^2)^{5/2}} dx$ is equal to

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)

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:

Indefinite Integration Questions (389)