JEE MAIN Indefinite Integration Previous Year Questions (PYQs) – Page 2 of 5

If $\displaystyle \int \frac{2x+5}{\sqrt{7-6x-x^{2}}},dx = A\sqrt{7-6x-x^{2}} + B\sin^{-1}!\left(\frac{x+3}{4}\right) + C$ (where $C$ is a constant of integration), then the ordered pair $(A,B)$ is equal to :

$ \text{The integral } \displaystyle \int \left[ \left(\frac{x}{2}\right)^{x} + \left(\frac{2}{x}\right)^{x} \right] \ln!\left(\frac{e x}{2}\right), dx \text{ is equal to:} $

The integral $\displaystyle \int \frac{x^{5}-x^{2}}{(x^{2}+3x+1)\,\tan^{-1}\!\left(x^{3}+\dfrac{1}{x^{2}}\right)}\,dx$ is equal to:

The integral $\displaystyle \int \frac{\sin^{2}x \cos^{2}x}{\left(\sin^{5}x + \cos^{3}x \sin^{2}x + \sin^{3}x \cos^{2}x + \cos^{5}x\right)^{2}},dx$ is equal to :

The integral $\displaystyle \int \cos(\log_e x)\,dx$ is equal to (where $C$ is a constant of integration):

If $\displaystyle \int \frac{\tan x}{1+\tan x+\tan^{2}x},dx = x - \frac{K}{\sqrt{A}}\tan^{-1}\left(\frac{K\tan x + 1}{\sqrt{A}}\right) + C,\ (C\ \text{is a constant of integration})$ then the ordered pair $(K,A)$ is equal to :

If $\displaystyle \int \frac{\tan x}{1+\tan x+\tan^2 x},dx = x - \frac{K}{\sqrt{A}}\tan^{-1}!\left(\frac{K\tan x + 1}{\sqrt{A}}\right) + C,\ (C\text{ is a constant of integration})$ then the ordered pair $(K,A)$ is equal to :

If $\displaystyle \int \frac{\sin^{2}x+\cos^{2}x}{\sqrt{\sin^{2}x\,\cos^{2}x}\;\sin(x-\theta)}\,dx = A\sqrt{\cos\theta\,\tan x-\sin\theta}\;+\;B\sqrt{\cos\theta-\sin\theta}\,\cot x + C,$ where $C$ is the integration constant, then $AB$ is equal to:

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$, if $\displaystyle \int \left( \left(\dfrac{x}{e}\right)^{2x} + \left(\dfrac{e}{x}\right)^{2x} \right) \log_e x \, dx = \dfrac{1}{\alpha} \left(\dfrac{x}{e}\right)^{\beta x} - \dfrac{1}{\gamma} \left(\dfrac{e}{x}\right)^{\delta x} + C$, where $e = \displaystyle \sum_{n=0}^{\infty} \dfrac{1}{n!}$ and $C$ is the constant of integration, then $\alpha + 2\beta + 3\gamma - 4\delta$ is equal to:

The integral $\displaystyle \int \frac{3x^{13}+2x^{11}}{(2x^{4}+3x^{2}+1)^{4}},dx$ is equal to: (where $C$ is a constant of integration)