Indefinite Integration Questions (389)

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

Let $I(x)=\displaystyle\int\frac{6}{\sin^2 x(1-\cot x)^2}\,dx$. If $I(0)=3$, then $I\!\left(\dfrac{\pi}{12}\right)$ is equal to

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

Question 25: Let \(I = \int \tan^6 x\, dx - \int \tan^4 x(\sec^2 x - 1)\, dx\). If \(I = \frac{\tan^5 x}{5} - \frac{\tan^3 x}{3} + A\tan x - x + D\) where \(A, B, C\) are constants with \(A = -\frac{1}{3}, B = 1, C = -1\), then \(A + B + C\) equals:

Let $\displaystyle\int\frac{2-\tan x}{3+\tan x}\,dx=\frac{1}{2}\!\left(\alpha x+\log_e|\beta\sin x+\gamma\cos x|\right)+C$, where $C$ is the constant of integration. Then $\alpha+\dfrac{\gamma}{\beta}$ is equal to:

Evaluate $\int \left( \frac{1}{8x+9} + e^{8x+9} \right) dx$

Find $\int [\cos(5+4x) + \sec^2(3-4x)] dx$

Find $\int \frac{x(\sin x + \cos x) + 1}{x} dx$

Evaluate $\int \frac{x^3 + x^2 + x + 1}{\sqrt{x^2 + 2x + 3}} dx$.

Find $\int \frac{1+x^2}{1+x^4} dx$

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

Evaluate $\int x \cos 2x \, dx$

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

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

The value of \(\int \frac{1-\ln x}{x^2} dx\) is

If $f(x) = \lim_{n \to \infty} \frac{\tan(1/n)\log(1/n)}{n}$, and $\int \frac{f(x)}{\sqrt{\sin^{11} x \cos x}} dx = g(x) + C$ (C being the constant of integration). Then:

$$\int \frac{x^4 - 2}{x^2\sqrt{x^4 + x^2 + 2}} dx =$$

Evaluate \(\int \left(\frac{x+2}{x+4}\right) e^x dx\)

A function $f(x)$ continuous on $\mathbb{R}$ and periodic with $2\pi$ satisfies $f(x) + (\sin x) f(x + \pi) = \sin^2 x$ then,

The integral \(\int x\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)dx\;(x>0)\) is equal to

If \(\int \dfrac{\cos^4 x\,dx}{(\sin^3 x)(\sin^5 x + \cos^5 x)^{3/5}}\), and the result involves constants \(A\) and \(B\) such that \(A = 5\) and \(B = \dfrac{2}{5}\), find \(AB\).

If \(f'(x) = 5^{x} \cdot 5^{f(x)}\), then \(k\) is:

If \(\int x^5 e^{-x^2} dx = g(x)e^{-x^2} + C\), where \(C\) is a constant of integration, then \(g(-1)\) is equal to

If \(\int 4x^3 e^{-4x^3} dx = \frac{1}{48}e^{-4x^3}f(x) + C\), where \(C\) is a constant of integration, then \(f(x)\) is equal to

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

We have \[I = \int \frac{\sin x}{\sin(x - \alpha)} dx\] If \(I = Ax + B\ln|\sin(x-\alpha)| + C\), then find the values of \(A\) and \(B\).

Given \(I = \int \dfrac{3x^{13} + 2x^{11}}{(2x^4 + 3x^2 + 1)^4}\,dx\). Then \(I\) equals:

[JEE Main 2019] \(\displaystyle\int\frac{2\sin x}{2+\sin 2x}\,dx\) equals (where \(C\) is a constant)

If \(\int f(x)dx = \Psi(x)\), then \(\int x^5 f(x^3)\,dx\) is equal to

Let \[I = \int \frac{dx}{(x^2 - 2x + 10)^2}\]If \(I = A\left[\tan^{-1}\left(\frac{x-1}{3}\right) + \frac{f(x)}{x^2 - 2x + 10}\right] + C\), then \(A\) and \(f(x)\) are:

If \(\int x^5 e^{-x^2} dx = g(x)e^{-x^2} + C\), where \(C\) is a constant of integration, then \(g(-1)\) is equal to:

Evaluate \(\int \frac{1+x}{\sqrt{1+x^4+x^2}} \cdot \frac{1}{1+x^4} dx\)

Given \(\int x^5 e^{-4x^3} dx = \frac{1}{48} e^{-4x^3} f(x) + C\)Then \(f(x)\) is equal to:

Evaluate $\int \frac{dx}{9-16x^2}$

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

The evaluation of \(\displaystyle\int e^{x-\frac{1}{x}}\cdot\frac{x^2+1}{x^2}\,dx\) is

Let \(x = f''(t)\cos t + f'(t)\sin t\) and \(y = -f''(t)\sin t + f'(t)\cos t\). Then \(\int\left[\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2\right]^{1/2} dt\) is equal to

If \(I = \int \dfrac{1}{2\sin x \cos x}\,dx + \dfrac{1}{2}\int \dfrac{\sqrt{\tan x}}{\sin x \cos x}\,dx\), and \(f(x) = \tan^2 x\), find \(f\!\left(\dfrac{\pi}{3}\right)\).

Evaluate \(\int \frac{(\sin x - 2\cos x)}{(2\sin x + \cos x)} dx\)