Find the integrals of the function: \(sin\, x\,\, sin\, 2x\,\, sin\, 3x\)
Solution and Explanation
It is known that, \(sin A\, sin B = \frac{1}{2}{cos(A-B)-cos(A+B)}\)
\(∴ ∫sin\, x\,\, sin\, 2x\,\, sin\, 3x. dx = ∫[sin x.\frac{1}{2}{cos(2x-3x)-cos(2x+3x)}]dx\)
\(= \frac{1}{2}∫(sinx\, cos(-x)-sinx\, cos\,5x)dx\)
\(= \frac{1}{2} ∫(sinx\,cosx-sinx\,cos5x)dx\)
\(= \frac{1}{2} ∫\frac{sin2x}{2}. dx -\frac{1}{2} ∫sin x\, cos5x. dx\)
\(= \frac{1}{4}[\frac{-cos2x}{2}]-\frac{1}{2} ∫{\frac{1}{2}sin(x+5x)+sin(x-5x)}dx\)
\(= \frac{-cos2x}{8}-\frac{1}{4} ∫(sin6x+sin(-4x))dx\)
\(=\frac{-cos2x}{8}-\frac{1}{4}[\frac{-cos6x}{3}+\frac{cos4x}{4}]+C\)
\(=\frac{-cos2x}{8}-\frac{1}{8}[\frac{-cos6x}{3}+\frac{cos4x}{2}]+C\)
\(= \frac{1}{8}[\frac{cos6x}{3}-\frac{cos4x}{2}-cos2x]+C\)
Top Questions on integral
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Rishika and Shivika were partners in a firm sharing profits and losses in the ratio of 3 : 2. Their Balance Sheet as at 31st March, 2024 stood as follows:
Balance Sheet of Rishika and Shivika as at 31st March, 2024Liabilities Amount (₹) Assets Amount (₹) Capitals: Equipment 45,00,000 Rishika – ₹30,00,000
Shivika – ₹20,00,00050,00,000 Investments 5,00,000 Shivika’s Husband’s Loan 5,00,000 Debtors 35,00,000 Creditors 40,00,000 Stock 8,00,000 Cash at Bank 2,00,000 Total 95,00,000 Total 95,00,000 The firm was dissolved on the above date and the following transactions took place:
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(ii) Investments were sold at a profit of 20% on its book value.
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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