Integrate the function: \(\frac {1}{\sqrt {8+3x-x^2}}\)
Solution and Explanation
8+3x-x2 can be written as 8 - (x2 - 3x + \(\frac 94\) - \(\frac 94\)).
Therefore,
8 - (x2- 3x + \(\frac 94\) - \(\frac 94\))
= \(\frac {41}{4}\) - (x-\(\frac 32\))2
⇒ \(∫\)\(\frac {1}{\sqrt {8+3x-x^2}}\ dx\) = \(∫\frac {1}{\sqrt {\frac {41}{4}-(x-\frac 32)^2}} dx\)
Let x-\(\frac 32\) = t
∴ dx = dt
⇒ \(∫\frac {1}{\sqrt {\frac {41}{4}-(x-\frac 32)^2}}\ dx\) = \(∫\frac {1}{\sqrt {(\frac {\sqrt {41}}{2})^2-t^2}} \ dt\)
= \(sin^{-1}(\frac {t}{\frac {\sqrt {41}}{2}})+ C\)
= \(sin^{-1}(\frac {x-\frac 32}{\frac {\sqrt {41}}{2}})+ C\)
= \(sin^{-1}(\frac {2x-3}{\sqrt {41}})+C\)
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Answer the following questions:
[(i)] Explain the structure of a mature embryo sac of a typical flowering plant.
[(ii)] How is triple fusion achieved in these plants?
OR
[(i)] Describe the changes in the ovary and the uterus as induced by the changes in the level of pituitary and ovarian hormones during menstrual cycle in a human female.
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Write a letter to the editor of a local newspaper expressing your concerns about the increasing “Pollution levels in your city”. You are an environmentalist, Radha/Rakesh, 46, Peak Colony, Haranagar. You may use the following cues along with your own ideas:
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Simar, Tanvi, and Umara were partners in a firm sharing profits and losses in the ratio of 5 : 6 : 9. On 31st March, 2024, their Balance Sheet was as follows:
Balance Sheet of Simar, Tanvi, and Umara as at 31st March, 2024
Liabilities Amount (₹) Assets Amount (₹) Capitals: Fixed Assets 25,00,000 Simar 13,00,000 Stock 10,00,000 Tanvi 12,00,000 Debtors 8,00,000 Umara 14,00,000 Cash 7,00,000 General Reserve 7,00,000 Profit and Loss A/c 2,00,000 Trade Payables 6,00,000 Total 52,00,000 Total 52,00,000 Umara died on 30th June, 2024. The partnership deed provided for the following on the death of a partner:
- Goodwill of the firm be valued at 3 years purchase of average profits for the last 5 years. The profits/losses for the previous four years were:
- 2022-23: ₹ 3,10,000 (loss)
- 2021-22: ₹ 3,00,000 (profit)
- 2020-21: ₹ 4,00,000 (profit)
- 2019-20: ₹ 2,50,000 (profit)
- Umara’s share of profit or loss till the date of her death was to be calculated on the basis of profit or loss for the year ended 31st March 2024.
- Calculate Goodwill of the firm.
- Pass the necessary journal entry for the treatment of goodwill on Umara’s death.
- Calculate Umara’s share in the profit or loss of the firm till the date of her death.
- Pass the necessary journal entry to record Umara’s share of profit or loss till the date of her death.
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- Goodwill of the firm be valued at 3 years purchase of average profits for the last 5 years. The profits/losses for the previous four years were:
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.
