Integrate the function: \(\frac{sin^8x-cos^8x}{1-2sin^2xcos^2x}\)
Solution and Explanation
\(\frac{sin^8x-cos^8x}{1-2sin^2xcos^2x}\)=\(\frac{(sin^4x+cos^4x)(sin^4x-cos^4x)}{sin^2x+cos^2x-sin^2xcos^2x-sin^2xcos^2x}\)
=\(\frac{(sin^4x+cos^4x)(sin^4x-cos^4x)}{sin^x(1-cos^2x)+cos^2x(1-sin^2x)}\)
=\(-\frac{(sin^4x+cos^4x)(cos^2x-sin^2x}{sin^4x+cos^4x}\)
=-cos2x
∴\(\frac{sin^8x-cos^8x}{1-2sin^2xcos^2x}\)=∫-cos2xdx=-sin2\(\frac{x}{2}\)+C
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Questions Asked in CBSE CLASS XII exam
Rupal, Shanu and Trisha were partners in a firm sharing profits and losses in the ratio of 4:3:1. Their Balance Sheet as at 31st March, 2024 was as follows:
(i) Trisha's share of profit was entirely taken by Shanu.
(ii) Fixed assets were found to be undervalued by Rs 2,40,000.
(iii) Stock was revalued at Rs 2,00,000.
(iv) Goodwill of the firm was valued at Rs 8,00,000 on Trisha's retirement.
(v) The total capital of the new firm was fixed at Rs 16,00,000 which was adjusted according to the new profit sharing ratio of the partners. For this necessary cash was paid off or brought in by the partners as the case may be.
Prepare Revaluation Account and Partners' Capital Accounts.- CBSE CLASS XII - 2025
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On the basis of the following hypothetical data, calculate the percentage change in Real Gross Domestic Product (GDP) in the year 2022 – 23, using 2020 – 21 as the base year.
Year Nominal GDP Nominal GDP (Adjusted to Base Year Price) 2020–21 3,000 5,000 2022–23 4,000 6,000 - CBSE CLASS XII - 2025
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- Altima Ltd. invited applications for 2,00,000 equity shares of ₹ 10 at a premium of ₹ 4 per share. Amount payable:
On application and allotment – ₹ 7 (incl. ₹ 1 premium)
On first and final call – Balance.
Applications received for 2,40,000 shares. 30,000 rejected. Manvi allotted 4,000 shares failed to pay first and final call. Her shares were forfeited. These were reissued at ₹ 4 per share fully paid-up.
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Concepts Used:
Definite Integral
Definite integral is an operation on functions which approximates the sum of the values (of the function) weighted by the length (or measure) of the intervals for which the function takes that value.
Definite integrals - Important Formulae Handbook
A real valued function being evaluated (integrated) over the closed interval [a, b] is written as :
\(\int_{a}^{b}f(x)dx\)
Definite integrals have a lot of applications. Its main application is that it is used to find out the area under the curve of a function, as shown below:
