Integrate the function: \(\frac {1}{\sqrt {(2-x)^2+1}}\)
Solution and Explanation
\(Let\ 2-x = t\)
\(⇒ -dx = dt\)
\(⇒\) \(∫\)\(\frac {1}{\sqrt {(2-x)^2+1}}\ dx\) = \(-∫\frac {1}{\sqrt {t^2+1} }dt\)
\(= -log\ |t+\sqrt {t^2+1}|+C\) \([∫\frac {1}{\sqrt {x^2+a^2} }dt = log\ |x+\sqrt {x^2+a^2}|]\)
\(=-log\ |2-x\sqrt {(2-x)^2+1}|+C\)
\(=log\ |\frac {1}{(2-x)+\sqrt {x^2-4x+5}}|+C\)
Top Questions on integral
- Evaluate the integral: \[ \int \frac{\sin(2x)}{\sin(x)} \, dx \]
- Evaluate the integral: \[ \int \sqrt{x^2 + 3x} \, dx \]
Let \( f : (0, \infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0 \), } \[ \int_0^a f(x) \, dx = f(a), \quad f(1) = 1, \quad f(16) = \frac{1}{8}, \quad \text{then } 16 - f^{-1}\left( \frac{1}{16} \right) \text{ is equal to:}\]
- Evaluate the following integral: \[ \int e^x \left[\frac{1}{1+x(1+x)^2}\right] dx \]
- Find the integral: \[ \int \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right) \, dx \]
Questions Asked in CBSE CLASS XII exam
Match Column-I with Column-II and choose the correct option:
- CBSE CLASS XII - 2025
- Colonial Cities: Urbanisation, Planning, and Architecture
Two statements are given below as Assertion and Reason (R). Read them carefully and choose the correct option.
Assertion : Harappa was a well-planned city.
Reason (R): It had a well-planned drainage system.- CBSE CLASS XII - 2025
- The Harappan Civilisation
Read the following source carefully and answer the questions that follow:
- CBSE CLASS XII - 2025
- Rulers and Buildings
Read the following source carefully and answer the questions that follow:
- CBSE CLASS XII - 2025
- Modern Indian History and Freedom Struggle
Read the given source carefully and answer the questions that follow:
- CBSE CLASS XII - 2025
- Modern Indian History and Freedom Struggle
Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
