Indefinite Integration Questions - JEE

If $\displaystyle\int e^x\!\left(\frac{x\sin^{-1}x}{\sqrt{1-x^2}} + \frac{\sin^{-1}x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2}\right)dx = g(x)+C$, where $C$ is the constant of integration, then $g\!\left(\dfrac{1}{2}\right)$ equals:

If $\displaystyle\int\frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}}\,dx=A\!\left(\dfrac{\alpha x-1}{\beta x+3}\right)^{\!B}+C$, where $C$ is the constant of integration, then the value of $\alpha+\beta+20AB$ is

If $\displaystyle\int\frac{dx}{a^2\sin^2 x+b^2\cos^2 x}=\frac{1}{12}\tan^{-1}(3\tan x)+\text{constant}$, then the maximum value of $a\sin x+b\cos x$ is:

If $\int \cosec^5 x\,dx = \alpha\cot x\cosec x\!\left(\cosec^2 x+\dfrac{3}{2}\right)+\beta\log_e\!\left|\tan\dfrac{x}{2}\right|+C$, where $\alpha,\beta\in\mathbb{R}$ and $C$ is the constant of integration, then the value of $8(\alpha+\beta)$ equals

Integer AnswerIf $\displaystyle\int x^2 e^{3x}\,dx = e^{ax}\left[bx^2 - cx + d\right]+c$, then $\dfrac{abc}{d}=$

Evaluate $\int e^{\tan^{-1} x} \left( \frac{1+x+x^2}{1+x^2} \right) dx$

Evaluate $$\int \frac{2 \sin 2x - \cos x}{6 - \cos^2 x - 4 \sin x} \, dx$$

If $\displaystyle\int \dfrac{3\tan\!\left(x - \dfrac{\pi}{4}\right)}{\cos^2 x\,\sqrt{\tan^3 x + \tan^2 x + \tan x}}\, dx = k\tan^{-1}\!\left(\sqrt{\tan x + 1 + \cot x}\right) + C$, then the value of $k$ is: [where $C$ is constant of integration.]

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

The derivative of $x^4 + x^{-5}$ is $-\left(4x^{-5} + 5x^{-6}\right)$. So, $$\int \frac{5x^3 + 4x^5}{\left(x^5 + x + 1\right)^2} dx =$$

If $f(x) = \lim_{n \to \infty} \left[2x + 4x^3 + ... + 2nx^{2n-1}\right]$ $(0 < x < 1)$ then $\int f(x)dx$ is equal to:

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \frac{x^2 - 3x - 6}{x^2 + 2x + 4}$. Then which of the following statements is (are) TRUE ?

$$\int_1 \frac{(2x^3 + 3x^2 - 1)\,\sqrt{4x^6 + 12x^5 + 9x^4 - 6x^3 - 9x^2 + 2}}{x(x+1)}\,dx =$$

$\int\sqrt{x}\tan\left(2\tan^{-1}\left(\frac{\sqrt{1+x+\sqrt{1}+x}-1}-\sqrt{1+\sqrt{x}-1}}{\sqrt{1+x+\sqrt{1}+x}-1}+\sqrt{1+\sqrt{x}-1}\right)\right)dx$ is equal to $ax^k + K\tan^{-1}\left(\frac{\sqrt{x}}{2}\right) + \frac{a}{\sqrt{1+\sqrt{x}}} + c$ then $a+b$ is equal to (where $a, b, k, a \in \mathbb{R}$)

If $y = f(x^2)$, $\forall x \in \mathbb{R}$ is equal to

Let $I(x)=\displaystyle\int\dfrac{x^2(x\sec^2x+\tan x)}{(x\tan x+1)^2}\,dx$. If $I(0)=0$, then $I\!\left(\dfrac{\pi}{4}\right)$ is equal to

Let $f(n, z) = \int \cos(nz) dx$, with $f(0, 0) = 0$. If the expression $\sum_{i=1}^{n} f(i, 1)$ simplifies to $\frac{\sin(a+b)}{\sin c}$, then the value of $\frac{a}{c}$ is (where $a > b$)

For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$, if $\displaystyle\int\!\left[\left(\frac{x}{e}\right)^{2x}+\left(\frac{e}{x}\right)^{2x}\right]\ln x\,dx=\dfrac{1}{\alpha}\!\left(\frac{x}{e}\right)^{\beta x}-\dfrac{1}{\gamma}\!\left(\frac{e}{x}\right)^{\delta x}+C$, then $\alpha+2\beta+3\gamma-4\delta$ is equal to

If $I_n = \int \cot^n x dx$, then $I_0 + I_1 + 2(I_2 + I_3 + ... + I_k) + I_9 + I_{10} =$

The integral $\int \frac{\sec^{3/2}\theta - \sec^{1/2}\theta}{2 + \tan^2\theta} \tan\theta d\theta$ is:

Let $I(x)=\displaystyle\int\frac{3\,dx}{(4z+6)\left(\sqrt{4x^2+8x+3}\right)}$ and $I(0)=\dfrac{\sqrt{3}}{4}+20$. If $I\!\left(\dfrac{1}{2}\right)=\dfrac{a\sqrt{2}}{b}+c$, where $a,b,c\in\mathbb{N}$, $\gcd(a,b)=1$, then $a+b+c$ is equal to

If $$\int \frac{dx}{x^2(x^n + 1)^{(n-1)/n}} = -[f(x)]^{1/n} + C$$, then $f(x)$ is

Evaluate: $\int \left( a^x \ln x + \ln a \cdot \ln\left(\frac{x}{e}\right) \right) dx$

If $\int \frac{\sin x}{\sin(x-a)} dx = Ax + B \log \sin(x-a) + C$, then the value of $(A, B)$ is:

If $\displaystyle\int\frac{x-1}{2x+1}\,dx = A\ln|2x+1|+Bx+C$, then

[JEE Main 2022] $\displaystyle\int\frac{dx}{1+3\sin^2 x+8\cos^2 x}$ equals (where $C$ is constant)

[JEE Main 2022] $\displaystyle\int\frac{\sqrt{\tan x}+\sqrt{\cot x}}{\sqrt{\sin x}\cdot\sqrt{\cos x}}\,dx$ equals (where $C$ is a constant)

[JEE Main 2022] $\displaystyle\int\sin^{-1}\!\frac{2x}{1+x^2}\,dx$ equals (for $|x|\le 1$)

[JEE Main 2019] $\displaystyle\int e^x\frac{1+\sin x}{1+\cos x}\,dx$ equals

[JEE Main 2022] $\displaystyle\int\frac{\ln x}{(1+x)^2}\,dx$ equals (where $C$ is constant)

[JEE Main 2020] $\displaystyle\int\frac{x+2}{\sqrt{x^2-1}}\,dx$ equals (where $C$ is a constant)

If the integral $\int \frac{5\tan x}{\tan x - 2}dx = x + a\ln|\sin x - 2\cos x| + k$, then a is equal to

$\int \frac{dx}{x(x^n+1)}$ is equal to

Write the given integration $I = \int \dfrac{dx}{x^{22}(x^7 - 6)}$ in the form involving substitution $p = 1 - \dfrac{6}{x^7}$. The answer is $\dfrac{1}{(42)(216)}\int \dfrac{1-p^3 - 3p + 3p^2}{p}\,dp$. Find the integer value of the denominator constant (i.e., $54432 = 42 \times 216 \times k$ gives $k$).

The function g(x) is

$\int \frac{\ln 2x}{4x}$ dx is equal to

Let $f(x) = \int_0^x e^{x-y} f'(y) dy - (x^2 - x + 1)e^x$. Find the number of roots of the equation $f(x) = 0$.

Indefinite Integration Questions (389)