Definite Integration Questions (1340)

\(\int \left\{\frac{(\log x - 1)}{(1+(\log x)^2)}\right\}^2 dx\) is equal to

Find $\int \frac{dx}{9x^2 + 4}$

If \( xf(x) = x + \int_1^x f(t)\,dt \), then find \( \sum_{k=1}^{10} f(e^k) \).

\(\int \frac{\sin^8 x - \cos^8 x}{1-2\sin^2 x\cos^2 x}dx\) is equal to

Evaluate the integral: \[ I = \int_{-1/\sqrt{3}}^{1/\sqrt{3}} \frac{\cos^{-1}\!\left(\dfrac{2x}{1+x^2}\right) + \tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)}{e^x + 1}\, dx \]

If $\int \frac{\sin^{3/2}\theta + \cos^{3/2}\theta}{\sin^8\theta \cos^9\theta} \sin(\theta + \alpha) d\theta = a\sqrt{\cos\alpha \tan\theta} + \sin\alpha + b\sqrt{\cos\alpha + \sin\alpha \cot\theta} + c$ then:

If m is a non-zero number and \(\int \frac{x^{5m-1}+2x^{4m-1}}{(x^{2m}+x^m+1)^3}dx = f(x)+C\), then f(x) is

Evaluate $\int \ln x \, dx$

\(\int \frac{dx}{\cos x + \sqrt{3}\sin x}\) equals

If \int_a^n \frac{x}{(a+n)e^{x(a+n)}} \, dx = \frac{2}{a+n}, then the value of \int_a^{2n} \frac{x}{(a+n)e^{x(a+n)}} \, dx, where a \neq -n, is

Evaluate $\int \frac{\sqrt{\tan x}}{\sin 2x} dx$

Evaluate $$\int \frac{\cos 2x}{\cos x} dx$$

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

Find $\int \frac{dx}{\sqrt{12x - 9x^2}}$

Evaluate the integral: \[I = \int \frac{dx}{x^2(x^4+1)^{3/4}}\]

Evaluate $\int e^x \sin x dx$

Evaluate: \[\int \frac{\sec^2 x - 2010}{\sin^{2010} x}\,dx\]Express the integral in simplest form and find the numerical value associated (given answer is 1.50).

If $\displaystyle\int \sqrt{\sec 2x - 1}\,dx = \alpha \log_e\left|\cos 2x + \beta + \sqrt{\cos 2x\left(1+\cos\dfrac{1}{\beta}x\right)}\right| + \text{constant}$, then $\beta - \alpha$ is equal to ______.

If \(f(x)\) is a differentiable function defined for all positive real numbers such that \(xf(x) = x + \displaystyle\int_1^x f(t)\,dt\), then the value of \(\displaystyle\sum_{k=1}^{10} f(e^k)\) is:

Evaluate $\int (\sin x + x \cos x) dx$

Given \(I = \displaystyle\int \frac{2x^3 - 1}{x^4 + x}\,dx\). Then \(I\) equals:

Anti-derivative of $$\frac{x - 1}{(x + 1)\sqrt{x^3 + x^2 + x}}$$ is:

If \(\int f(x)\,dx = \Psi(x)\), then \(\int x^5 f(x^3)\,dx\) equals:

Given\[ I = \int \frac{\sin\left(\dfrac{5x}{2}\right)}{\sin\left(\dfrac{x}{2}\right)}\, dx \]Evaluate \(I\).

Evaluate the integral: \[I = \int \frac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3} dx\]

Let \( I_n = \displaystyle\int_0^{\pi/2} \dfrac{\sin^2 nx}{\sin x} \, dx \) and \( I_{n+1} = \displaystyle\int_0^{\pi/2} \dfrac{\sin^2(n+1)x}{\sin x} \, dx \). Then \( I_{n+1} - I_n \) equals:

For $x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$, if $y(x)=\displaystyle\int\dfrac{\csc x+\sin x}{\csc x\sec x+\tan x\sin^2x}\,dx$ and $\lim_{x\to(\pi/2)^-}y(x)=0$, then $y\left(\dfrac{\pi}{4}\right)$ is equal to

If \(\int \frac{1 - x^7}{x(1 + x^7)} dx = P \log|x| + Q \log|x^7 + 1| + C\), then:

If \(I = \int \frac{dx}{(x-1)^4(x+2)^3} = k\sqrt[4]{x+2} + C\), then \(k\) is equal to:

Evaluate: \(I = \int \frac{\sin^8 x - \cos^8 x}{1 - 2\sin^2 x \cos^2 x} dx\)

Evaluate: \(\int \frac{\sin 2x}{(3 + 4\cos x)^3} dx\)

If \int \frac{\log(x + \sqrt{1 + x^2})}{1 + x}dx = g \circ f(x) + \text{constant}, then (JEE Main 2016)

Find \(\int \frac{3x+1}{(x+1)^3} dx\)

Evaluate \(\int \frac{(x+2)\, dx}{(x^2 + 3x + 3)\sqrt{x+1}}\)

\(\int \frac{\tan x}{\sin x \cos x} dx\) is equal to [IIT - 1998]

\(\int \frac{x+1}{(x^2+1)^{3/2}} dx\) equals

Find \(\int \frac{dx}{(x-p)\sqrt{(x-p)(x-q)}}\)

The value of $\displaystyle \int \frac{dx}{\sqrt{a^2-x^2}}$ is equal to

\(\int \frac{\sec x}{(\sec x + \tan x)^{9/2}} dx\) equals (for some arbitrary constant K) [IIT - 2012]

Evaluate \int x^3 \sqrt[4]{1 + x^{1/3}} \, dx

If \(f(x) = \sqrt{\frac{x + 2}{2x + 3}}\), then evaluate \(\int \frac{f(x)}{x^{1/2}} dx\)

\(\int \frac{\sin 2x \cos 2x}{(\sin^5 x + \cos^3 x \sin^2 x + \sin^3 x \cos^2 x + \cos^5 x)^2} dx\) is equal to (JEE Main 2018)

\(\int \frac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}\,dx\) is equal to

If \(\lim_{x \to 0} \left(\sum_{r=1}^n \cos\dfrac{r\pi}{2n}\right)\left(\sum_{r=1}^n \cos^2\dfrac{r\pi}{2n}\right)\left(\sum_{r=1}^n \cos^3\dfrac{r\pi}{2n}\right)\left(\sum_{r=1}^n \cos^4\dfrac{r\pi}{2n}\right)\left(\dfrac{1}{e^n - 1}\right)^4 = \dfrac{k}{\pi^2}\), then find the value of \(\dfrac{1}{k}\).

\(\int x^m x^n - 1 + \frac{1}{x^n} - \frac{mx}{n} + \frac{nx}{m} - \frac{1}{2} - \frac{n}{2m} + \frac{m}{n} dx\) is equal to

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

If \(\int f(x)\,dx = f(x) + C\), then \(\int [f(x)]^2\,dx\) is

The value of \(\int \frac{f(x)\phi'(x) - \phi(x)f'(x)}{[f(x)\phi(x)-1]^2 - f(x)\phi(x) - 1} dx\) is

Evaluate \(\int \frac{\ln x}{x(1 + \ln x)} dx\)

The value of $\displaystyle \int \frac{e^x + x^4 + 2}{(1+x^2)^{5/2}} dx$ is equal to