Indefinite Integration Questions - JEE

Evaluate: $\int \frac{\cos ec^2 x - 2005}{\cot x + \tan x} dx$

Let \int x sin x dx = g(x) + C , where C is the constant of integration. If 3 8 (g ( \pi 2 ) + g ( ′ \pi 2 )) = \alpha\pi 3 + \beta\pi 2 + \gamma, \alpha, \beta, \gamma \in Z , then \alpha + \beta - \gamma equals :

$\int \frac{x^2 - 1}{(x^4 + 3x^2 + 1)\tan^{-1}\left(x + \frac{1}{x}\right)} dx$ is equal to

The integral $\displaystyle\int\dfrac{(x^8-x^2)\,dx}{(x^{12}+3x^6+1)\tan^{-1}\!\left(x^3+\dfrac{1}{x^3}\right)}$ is equal to:

Primitive of $\dfrac{3x+1}{(x+1)^2\sqrt{x}}$ w.r.t. $x$ is

Evaluate\[ I = \int \sec^{2/3} x\, \csc^{4/3} x\, dx \]

78. $\int |x| dx$ is equal to

Let $\displaystyle\int x^3\sin x\,dx = g(x)+C$, where $C$ is the constant of integration. If $8\!\left(g\!\left(\dfrac{\pi}{2}\right)+g'\!\left(\dfrac{\pi}{2}\right)\right) = \alpha\pi^3+\beta\pi^2+\gamma$, $\alpha,\beta,\gamma\in\mathbb{Z}$, then $\alpha+\beta-\gamma$ equals:

∫ dx/√(1-tan²x) = λ sin⁻¹(λ sinx) + C, then λ = ?

Let $f(x)=\displaystyle\int\dfrac{dx}{(3+4x^2)\sqrt{4-3x^2}}$, $|x|<\dfrac{2}{\sqrt{3}}$. If $f(0)=0$ and $f(1)=\dfrac{1}{\alpha}\tan^{-1}\!\left(\dfrac{\alpha}{\beta}\right)$, $\alpha,\beta>0$, then $\alpha^2+\beta^2$ is equal to _______.

Assertion (A): The function $F(x)$ (an indefinite integral of $\sin 2x$) satisfies $F(x + \pi) = F(x)$ for all real $x$.Reason (R): $\sin 2(x + \pi) = \sin 2x$ for all real $x$.

Evaluate $\int \frac{e^x(1+x)}{\cos^2(e^x x)} dx$

Evaluate $\int \sqrt{x^2 + 4x + 1}\,dx$

$\int (\sin 2x - \cos 2x) \, dx = \frac{1}{2}\sin(2x - a) + b$, then

79. $\int \frac{(1-x^2)^{3/2}}{x^2} dx$ is equal to

Let $f(x)=\displaystyle\int\frac{dx}{x^{2/3}+2x^{1/2}}$ be such that $f(0)=-26+24\log_e 2$. If $f(1)=a+b\log_e 3$, where $a,b\in\mathbb{Z}$, then $a+b$ is equal to:

The value of \ \int \cos^{\frac{1}{2}} x \cdot \sin^3 x \, dx\ is _______.

[JEE Main 2021] $\displaystyle\int x^5\sqrt{1+x^3}\,dx$ equals (where $C$ is a constant)

$\int \frac{e^{\tan^{-1} x}}{1+x^2}\,dx$ is equal to

Evaluate: \int \frac{\cot x}{(\cos x)^{2005}} dx

The integral \int \frac{(2x - 1) \cos\sqrt{(2x-1)^2 + 5}}{4x^2 - 4x + 6} dx is equal to (where C is a constant of integration) (JEE Main 2021)

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

The integral \int (x\sin x + \cos x)dx is equal to

Evaluate $\int \frac{dx}{(x+3)^{15/16}(x-4)^{17/16}}$

$\displaystyle\int \frac{1}{x^2\sqrt{1-x^2}}\,dx$ equals

If $\int (e^{2x} + 2e^x \cos x - e^{-x}(e^x + e^{-x})) dx = g(x)e^{(e^x + e^{-x})} + c$, where $c$ is a constant of integration, then $g(0)$ is equal to

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

$\int \frac{2^x + 3^x}{5^x}$ dx is equal to

Let $f(x) = \int \frac{x^2 \, dx}{(1+x^2)(1+\sqrt{1+x^2})}$ and $f(0) = 0$, then the value of $f(1)$ is

$\int x^x(1 + \log|x|) \, dx$ is equal to

$\int \frac{x^2}{x^4 + x^2 + 1} dx$ is equal to

85. If $\int f(x) dx = F(x)$, then $\int x^3 f(x^2) dx$ is equal to

Evaluate $\int \frac{1}{\cos x} \cdot \sin 2x \cdot \cos 2x \, dx$

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

If f(y) = eʸ, g(y) = y, y > 0 and F(t) = ∫₀¹ f(t − y)g(y)dy, then F(t) is

Suppose that F(x) is an antiderivative of f(x) = (sin x)/(x) − (3 sin 2x)/(1 + x²), where x > 0, then ∫ dx can be

Evaluate $\int_3^5 x^5 \sqrt[3]{1 + 3x^4} \, dx$

If $f(x) = A \sin(x/2) + B$, $f'(\pi/2) = 2$, and $\sqrt{2}\int_0^1 f(x)dx = \frac{2A}{\pi}$, then the constants A and B are

If $f\left(\dfrac{x-4}{x+2}\right) = 2x + 1$, $(x \in \mathbb{R} - \{1, -2\})$, then $\int f(x)\, dx$ is equal to (where $C$ is a constant of integration):

If $I_n = \int \tan^n x\, dx$, then $I_4 + I_6 = a\tan^5 x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to:

The number of solution(s) of the equation f(x) = x^3 in [0, 2\pi] be:

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

If $I = \int \frac{x^2 - 1}{x^3(2x^4 - 2x^2 + 1)} dx$ is equal to:

[JEE Main 2021] If $\displaystyle\int f(x)\,dx=\psi(x)$, then $\displaystyle\int x^5 f(x^3)\,dx$ equals

$\displaystyle\int\frac{\sin(x-a)}{\sin(x+a)}\,dx$ equals

$\int \frac{\sec 5x}{\sin^3 x} dx$ equals to:

Evaluate: $\int \frac{1 - x^2}{1 + x^4} dx$

If $\int \frac{(\sqrt{x})^5}{(1-\sqrt{x})^7} \cdot \frac{\sqrt{x}}{(1-\sqrt{x})^k} dx = a \ln \left|\frac{\sqrt{x}}{1-\sqrt{x}^k}\right| + c$, then the values of $a$ and $k$ are

$\int \frac{g(x)}{f(x)}$ dx is equal to

If $f(x) = \cos x - \cos 2x + \cos 3x - \ldots \infty$, then $\int f(x) dx$ is equal to

Indefinite Integration Questions (389)