Find the integrals of the function: \(sin^3 (2x+1)\)
Solution and Explanation
Let \(I = ∫sin^3(2x+1)\)
\(⇒ ∫sin^3(2x+1)dx = ∫sin^2(2x+1).sin(2x+1)dx\)
\(= ∫(1-cos^2(2x+1)sin(2x+1)dx\)
Let \(cos(2x+1)=t\)
\(⇒ -2sin(2x+1)dx = dt\)
\(⇒ sin(2x+1)dx = \frac{-dt}{2}\)
\(⇒ I = \frac{-1}{2} ∫(1-t^2)dt\)
\(=-\frac{1}{2}[t-\frac{t^3}{3}]\)
\(=\frac{-1}{2}{cos(2x+1)-\frac{cos^3(2x+1)}{3}}\)
\(=\frac{-cos(2x+1)}{2}+\frac{cos^3(2x+1)}{6}+C\)
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- Pulkit and Ravinder were partners in a firm sharing profits and losses in the ratio of 3 : 2. Sikander was admitted as a new partner for \( \frac{1}{5} \)th share in the profits of the firm. Pulkit, Ravinder and Sikander decided to share future profits in the ratio of 2 : 2 : 1. Sikander brought Rs 5,00,000 as his capital and Rs 10,00,000 as his share of premium for goodwill. The amount of premium for goodwill that will be credited to the old partners' capital accounts will be :
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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:
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(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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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C