What is the integral of cos^3(x)tan^4(x)?

Thread starter baher
Start date Jan 25, 2012
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Integral

In summary, the first step in finding the integral of cos^3(x)tan^4(x) is to use the power reduction formula to rewrite the integral as a product of two trigonometric functions. Then, the next step is to expand the product and simplify using trigonometric identities. The final step is to integrate the resulting polynomial expression using the power rule for integration to obtain the final answer in terms of x.

Jan 25, 2012

baher

i spent a lot of time solving this question but i can't solve it

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Jan 25, 2012

sponsoredwalk

cos³(x)tan⁴(x) = cos³(x)[tan²(x)]²

Think of a way to re-express tan²(x), expand & the rest should follow...

1. What is the first step in finding the integral of cos^3(x)tan^4(x)?

The first step is to use the power reduction formula to rewrite the integral as a product of two trigonometric functions.

2. How do you use the power reduction formula to rewrite the integral?

The power reduction formula states that cos^2(x) = 1/2(1+cos(2x)), which can be used to rewrite cos^3(x) as cos(x)(1+cos(2x)). Similarly, tan^2(x) = sec^2(x)-1, which can be used to rewrite tan^4(x) as (sec^2(x)-1)^2.

3. What is the next step after rewriting the integral using the power reduction formula?

The next step is to expand the product of the two trigonometric functions and use the trigonometric identities to simplify the expression.

4. How do you simplify the expanded expression?

Use the identities cos(x)cos(2x) = 1/2(cos(3x)+cos(x)) and sin(x)cos(2x) = 1/2(sin(3x)+sin(x)) to simplify the expanded expression. This will result in a polynomial expression.