Indefinite Integration Questions - JEE

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

Let $f(x)=\displaystyle\int\frac{(2-x^2)\cdot e^x}{(\sqrt{1+x})(1-x)^{3/2}}\,dx$. If $f(0)=0$, then $f\!\left(\dfrac{1}{2}\right)$ is equal to:

$\int (\tan x + \cot x) dx$ is equal to

Let $f(t)=\displaystyle\int\left(\frac{1-\sin(\log_e t)}{1-\cos(\log_e t)}\right)dt$, $t>1$. If $f(e^{\pi/2})=-e^{\pi/2}$ and $f(e^{\pi/4})=\alpha e^{\pi/4}$, then $\alpha$ equals

If $\int \frac{1+3\tan x(\tan x + \sec x)}{\tan x} dx = a \log \left|\cos \frac{x}{2} + \sin \frac{x}{2}\right| + C$ where $0 , then \(a$ is equal to

Given $f(x) = \int \dfrac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} dx$. If $f(0) = 0$, then find the value of $f(1)$.

If $\int\left[\ln\left(\frac{\cos 2\theta}{1+\sin 2\theta}\right) + \ln\left(\frac{1+\sin 2\theta}{1-\sin 2\theta}\right)^{\cos^2\theta}\right]d\theta$ is equal to $\frac{1}{a}\sin 2\theta\ln\left|\frac{\cos\theta + \sin\theta}{\cos\theta - \sin\theta}\right| + b\ln|\cos 2\theta| + c$ where $a, b \in \mathbb{R} - \{0\}$ & $c$ is integration constant such that $\cos\theta > \sin\theta > 0$ then $(a+b)$ is

$\int \tan x dx$ is equal to

Let $f$ & $g$ be differentiable function for all $x \in \mathbb{R}$ & have the following properties (i) $f'(x) = f(x) - g(x)$ (ii) $g'(x) = g(x) - f(x)$ (iii) $f(0) = 5$ (iv) $g(0) = 1$ Then the value of $|f(\ln 2) + g(\ln 3)|$ is equal to

If $\int\frac{x^3 + x + 1}{x^4 + x^2 + 1}dx = A_1\ln(x^2 + x + 1) + A_2\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right) + A_3\tan^{-1}\left(\frac{2x-1}{\sqrt{3}}\right) + A_4\tan^{-1}\left(\frac{2x^2+1}{\sqrt{3}}\right) + c$ then the value of $(A_1 + A_2 + A_3 + A_4)$ is

Let $\int\frac{\ln\left(x + \sqrt{1+x^2}\right)}{\sqrt{1+x^2}}dx = fog(x) + c$, where $f(x) = \frac{x^2}{2}$ and $g$ are some functions and $c$ is an arbitrary constant. If $\int f(x)g(x)dx = ax^3g(x) + b\left(1+x^2\right)^{3/2} + c\left(1+x^2\right)^{1/2} + d$, then $\left(\frac{1}{a+b+c}\right)$ is equal to

If $\int\frac{(\cos x - \sin x + 1 - x)}{e^x + \sin x + x}dx = \ln(f(x)) + g(x) + c$ where $c$ is the constant of integration & $f(x)$ is positive, then $\frac{f(x) + g(x)}{e^x + \sin x}$ is

Indefinite Integration Questions (389)