Integrate the function: \(\frac {1}{\sqrt {(2-x)^2+1}}\)
Solution and Explanation
\(Let\ 2-x = t\)
\(⇒ -dx = dt\)
\(⇒\) \(∫\)\(\frac {1}{\sqrt {(2-x)^2+1}}\ dx\) = \(-∫\frac {1}{\sqrt {t^2+1} }dt\)
\(= -log\ |t+\sqrt {t^2+1}|+C\) \([∫\frac {1}{\sqrt {x^2+a^2} }dt = log\ |x+\sqrt {x^2+a^2}|]\)
\(=-log\ |2-x\sqrt {(2-x)^2+1}|+C\)
\(=log\ |\frac {1}{(2-x)+\sqrt {x^2-4x+5}}|+C\)
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Bittu and Chintu were partners in a firm sharing profit and losses in the ratio of 4 : 3. Their Balance Sheet as at 31st March, 2024 was as follows:
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Vibha was appointed to look after dissolution work for which she was allowed a remuneration of ₹ 16,000. Vibha agreed to bear the dissolution expenses. Actual dissolution expenses ₹ 15,000 were paid by Vibha.
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A machine which was not recorded in the books was taken over by Vibha at ₹ 23,000, whereas its expected value was ₹ 28,000.
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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.
