Definite Integration Questions (1340)

If \(x\sin(f(x)) + \displaystyle\int_0^x \sin(f(t))\,dt = (x+2)\sin(f(x)) + \displaystyle\int_0^x t\sin(f(t))\,dt\), then find the value of \(f'(x)\cot(f(x)) + \dfrac{3}{1+x}\).

If

$\int_0^\pi |\sin x| \, dx$

$\int_0^{2\pi} |2\sin x| \, dx$

Evaluate $I = \displaystyle\int_{e^{\pi/6}}^{e^{\pi/2}} \frac{\sin(\ln(\sin(\ln x)))\cdot\cos(\ln x)}{x\sin(\ln x)}\,dx$. Find $\cos^{-1}(I+1)$.

Evaluate: \int \frac{\cos 2x - 2x \operatorname{cosec}^2 2005}{\sin^2 x \cos x} dx

If \[f(x) = \begin{vmatrix} x^2+4x-6 & 2x^2-4x-10 & 3x^2+2x-16 \\ x^2+2 & 2x-2 & 3x+1 \\ 1 & 2 & 3 \end{vmatrix}\] then find the value of \[\int_{-3}^{1} f(x)dx\]

Evaluate: \int x^5 \sqrt{1+x^3} dx

Evaluate \int_{10\pi + \frac{\pi}{6}}^{10\pi + \frac{\pi}{3}} (\sin x + \cos x) \, dx

∫ \frac{1-x^7}{x(1+x^7)} dx \text{ equals -}

If \(g(1) = g(2)\), then \(\int_1^2 \frac{[f\{g(x)\}]^{-1} f'\{g(x)\} g'(x)}{f^2(x)} dx\) is equal to

Evaluate: \ 5\int \frac{dx}{\sqrt{1+x} - \sqrt[3]{1+x}}

Suppose f is continuous and satisfies f(x) + f(-x) = x^2 then find the value of \int_{-1}^{1} f(x) dx

Consider f(x) = \frac{x^2}{1+x^3}; g(t) = \int f(t)dt. If g(1) = 0 then g(x) equals -

If $y = f(x) = \frac{3x}{2}$ and $g(x) = f^{-1}(x)$, find $g(1)$ where the curve $y = f^{-1}(x)$ passes through $\left(1, -\frac{2}{3}\right)$

The integral \int \frac{(x+1)^3 + x + x^2 dx}{1 + x} is equal to

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

If ∫\frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \ln |9e^{2x} - 4| + C, then

The value of the integral ∫π/4-π/4 \left( x + \frac{\sin^2 x}{\lfloor x \rfloor + 2} \right) dx is (where [x] denotes the greatest integer less than or equal to x)

Consider the functions f(x) and g(x), both defined from \mathbb{R} \to \mathbb{R}:f(x) = \frac{x^3}{3} + 1 - x \int_{0}^{x} g(t) dtg(x) = x - \int_{0}^{1} f(t) dtThe minimum value of f(x) is:

For each positive integer \(n\), a function \(f_n\) is defined on \([0,1]\) as: \[f_n(x) = \begin{cases} 0 & \text{if } x=0 \\ \sin\frac{\pi}{2n} & \text{if } 0

Evaluate \(\displaystyle\int_0^{\pi/2}\frac{dx}{1+\sqrt{\tan x}}\) [JEE Main 2022]

Let F(x) be an indefinite integral of sin2x.Statement-1 : The function F(x) satisfies F(x + π) = F(x) for all real x.becauseStatement-2 : sin2(x + π) = sin2x for all real x.

Let $I(x) = \int \sqrt{\frac{x+7}{x}} dx$ and $I(9) = 12 + 7 \log_e 7$. If $I(1) = \alpha + 7 \log_e (1 + 2\sqrt{2})$, then $\alpha^4$ is equal to________.

Evaluate $\int_{-3}^{5} e^{\{x\}} dx$, where $\{.\}$ denotes the fractional part function.

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

Evaluate $\int_{\alpha}^{\beta} \frac{dx}{\sqrt{(x-\alpha)(\beta-x)}}, \beta > \alpha$

Evaluate $\int e^{ax} \sin bx dx$ and $\int e^{ax} \cos bx dx$

The value of $\int_{0}^{1} \frac{x \tan^{-1} x}{(1+x^{2})^{3/2}} dx$ is

Evaluate $\int \frac{dx}{3x^2+6x+15}$

Evaluate $\int_{2}^{3} \frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}} dx$

If $\int_0^a f(x)dx = k$, then the value of $\int_0^a f(kx)dx$ is equal to

Let \( I = \int_0^1 x(1-x)^{99} \, dx \). Find the value of \(I\).

Let $f: \mathbb{R} \to \mathbb{R}$ is a function defined as $f(x) = \begin{cases} |x| & [x] \text{ is odd} \\ x - [x+1] & [x] \text{ is even} \end{cases}$ where $[\cdot]$ denotes the greatest integer function, then $\int_{-1}^2 f(x)dx$ is equal to

Evaluate $\int \sin^{-1} x \, dx$

Evaluate $\int e^{\tan x} (\sec x - \sin x) dx$

If $f$ is a continuous function and $\phi(x) = \int_{0}^{x} \left( (3t+4) \int_{t}^{3} f(u) du \right) dt$ and $\int_{0}^{3} f(x) dx = 3$, then:

Let F(x) be an indefinite integral of sin^2 x.Statement-1: The function F(x) satisfies F(x + π) = F(x) for all real x.becauseStatement-2: sin^2(x + π) = sin^2 x for all real x.

Find $\int \frac{dx}{\sqrt{a^2 - x^2}}$

Let I = \int \frac{e^x}{e^{4x} + e^{2x} + 1} dx, J = \int \frac{e^{-x}}{e^{-4x} + e^{-2x} + 1} dx. Then, for an arbitrary constant c, the value of J - I equals

Evaluate $\int_{-1}^{2} x[x] dx$

Find $\int \sqrt{3 - 2x - x^2} dx$

Illustration 1: From the given graph of $y = f(x)$, find the values of the following definite integrals:$\int_a^b f(x) dx$$\int_b^a f(x) dx$$\int_e^c f(x) dx$$\int_a^d f(x) dx + \int_b^e f(x) dx$

Find $\int \sqrt{x^2 + 2x + 5} dx$

Evaluate $\int_{-\pi}^{\pi} \frac{x \sin x}{e^x + 1} dx$

Evaluate $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} \cdot \frac{1}{x} dx$

Evaluate $\int_{0}^{\infty} (\cot^{-1} x)^2 dx$

Find $\int \frac{2x^3 dx}{1+x^2}$

Find $\int \sqrt{\frac{x-3}{5-x}}dx$

Evaluate $\int \sqrt{x^2 - 3x^6} dx (x > 0)$