((^(-1) x))/(1 + x^2) dx = Option 1: -(^(-1) x) + C Option 2: (^(-1) x) + C Option 3: (^(-1) x) + C Option 4: -(^(-1) x) + C

Answer: -(^(-1) x) + C. Solution: Substitute t = ^(-1) x, so dt = dx/(1+x^2). The integral becomes t dt = - t + C = -(^(-1) x) + C.

Calculus

Integrals

Easy

$\int \frac{\sin(\tan^{-1} x)}{1 + x^2} dx =$

$-\cos(\tan^{-1} x) + C$

$\cos(\tan^{-1} x) + C$

$\tan(\cos^{-1} x) + C$

$-\tan(\cos^{-1} x) + C$

✅ Correct Option: 1

Substitute $t = \tan^{-1} x$ , so $dt = \frac{dx}{1+x^2}$ . The integral becomes $\int \sin t\, dt = -\cos t + C = -\cos(\tan^{-1} x) + C$ .

2022: 23 Aug Shift 1

$\int \sqrt{x^2 - 4x + 5} \, dx =$

2026: 21 May Shift 2

Let $[t]$ denotes the greatest integer $\le t$ , then the value of $\left[\displaystyle\int_{\pi}^{3\pi/2} x|\sin x|\,dx\right]$ is:

2022: 30 Aug Shift 1

$\int x\sqrt{x + 2} \, dx$ is equal to :

2025: 15 May Shift 2

If $I = \int \frac{x}{x - \sqrt{x^2 - 4}} dx = \alpha x^3 + \beta(x^2 - 4)^{\frac{3}{2}} + \gamma$ , where $\gamma$ is constant of integration, then

2026: 30 May Shift 2

Match List-I with List-II

Choose the correct answer from the options given below:

2026: 22 May Shift 1

$\int \frac{\sqrt{9+(\log x)^2}}{x} dx$ is equal to: (where $C$ is an arbitrary constant and consider $\log_e x = \log x$ )