# Integrate 4cos(x/2)cos(x)sin(21x/2) Homework

• Tanishq Nandan
In summary, the conversation discusses the integration of 4cos(x/2).cos(x).sin(21x/2) and the use of trigonometric identities to simplify the product of trigonometric terms. The speaker suggests using product-to-sum identities to transform the expression into a simpler form for integration.
Tanishq Nandan

## Homework Statement

Integrate: 4cos(x/2).cos(x).sin(21x/2)

## Homework Equations

▪sin(A+B)=sinAcosB+sinBcosA
▪2sinAcosA=sin2A
And obviously,
▪Integration of sinx is (-cosx)
▪Integration of cosx is (sinx)

## The Attempt at a Solution

○I multiplied the numerator and denominator with sin(x/2)
○The numerator simplified to sin(2x)sin(21x/2) and in the denominator,we have sin(x/2).
○Now,I expanded sin(21x/2) by breaking 21x/2 into 10x and x/2 and using the above specified formula,which,on simplifying gives two terms.
○Now,the first term comes out to be sin2xcos10x,which can be integrated easily (well,maybe not one step but at least it can be done),but the second term is: sin(2x).sin(10x).cot(x/2)
How to integrate this term,it just eludes me..

Well I wouldn't recommend this approach. You should instead consider the product-to-sum identities in order to simplify the product of trigonometric terms into a simple sum of individual sine and cosine terms. For example, ##2 \cos \theta \cos \phi = \cos (\theta + \phi) + \cos (\theta - \phi)##.

Greg Bernhardt
Oh..ok,got it.Thanks

## 1. What is the formula for integrating 4cos(x/2)cos(x)sin(21x/2)?

The formula for integrating 4cos(x/2)cos(x)sin(21x/2) is ∫4cos(x/2)cos(x)sin(21x/2) dx = ∫2cos(21x/2)cos(x) dx.

## 2. How do I solve this integral?

To solve this integral, you can use the trigonometric identity cos(a)cos(b) = 1/2(cos(a+b) + cos(a-b)). This will simplify the integral to ∫cos(21x/2 + x) + cos(21x/2 - x) dx. Then, you can use substitution or integration by parts to solve the integral.

## 3. Can I use the power rule to integrate this expression?

No, you cannot use the power rule to integrate this expression. The power rule only applies to integrals of the form ∫xn dx, where n is a constant. This integral involves trigonometric functions, so the power rule cannot be applied.

## 4. Is there a shortcut or trick to solving this integral?

There is no specific shortcut or trick to solving this integral. However, using the trigonometric identity mentioned in question 2 can simplify the integral and make it easier to solve.

## 5. What are the possible ways to check my answer for this integral?

One way to check your answer is to take the derivative of your solution and see if it matches the original expression. Another way is to use a graphing calculator to graph both the original expression and your solution and see if they match. You can also use an online integral calculator to check your answer.

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