How to find antiderivative of product

In summary, donjt81 was trying to find the anti derivative of x^5 + tan(2x)sec(2x)dx. After looking at the equation, he realized that he needed to use u-substitution twice to get the correct answer.
  • #1
donjt81
71
0
problem: find the anti derivative of x^5 + tan(2x)sec(2x)dx

how do you find the anti derivative of the second half of that problem tan(2x)sec(2x)
 
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  • #2
donjt81 said:
problem: find the anti derivative of x^5 + tan(2x)sec(2x)dx

how do you find the anti derivative of the second half of that problem tan(2x)sec(2x)

Does the function y=sec(u)tan(u) look familiar at all? Perhaps as the derivative of some common elementary function?
 
  • #3
yeah, you are going to have to use u-substitution twice. I don't know if I am correct or not, but I changed everything to sin and cos before anything and manipulated it that way. I then let u = cosx and du = -sinx. For the second u-substitution, I let v=u²-1 and (1/2)dv = udu. I hope that helps and does not confuse you and more.
 
  • #4
prace said:
yeah, you are going to have to use u-substitution twice. I don't know if I am correct or not, but I changed everything to sin and cos before anything and manipulated it that way. I then let u = cosx and du = -sinx. For the second u-substitution, I let v=u²-1 and (1/2)dv = udu. I hope that helps and does not confuse you and more.

What are you doing? What is the derivative of secant?
 
  • #5
oh yeah... I see that the deriveitive of secx = secx tanx... Hmm... looks like I went a long.. long long long way around it.. haha. Sorry about that donjt81, I hope I didn't lead you too far the wrong way. I think it comes out to the same answer. Does it? Well, I guess we can both learn a lesson, or at least myself. And that is to really look at what's in front of you with the equation before just jumping into it. I just jumped right in and started to manipulate it, when I could have just taken a minute to think about what was really there and solved it much quicker and easier.
 

Related to How to find antiderivative of product

1. How do I find the antiderivative of a product?

The first step in finding the antiderivative of a product is to use the product rule, which states that the antiderivative of a product is equal to the first function multiplied by the antiderivative of the second function, plus the antiderivative of the first function multiplied by the second function. This can be written as ∫(f(x)g(x))dx = ∫f(x)dx * g(x) + ∫g(x)dx * f(x).

2. What is the difference between a derivative and an antiderivative?

A derivative is a measure of how a function changes over time or space, while an antiderivative is the opposite process of finding the original function from its derivative. In other words, a derivative tells us the rate of change of a function, while an antiderivative tells us the original function that would result in that rate of change.

3. Can I use the power rule to find the antiderivative of a product?

No, the power rule can only be used to find the derivative of a function, not the antiderivative. To find the antiderivative of a product, you will need to use the product rule or other integration techniques such as substitution or integration by parts.

4. Are there any general rules for finding the antiderivative of a product?

Yes, in addition to the product rule, there are other general rules for finding the antiderivative of a product. These include the quotient rule, chain rule, and u-substitution. It is important to know and understand these rules in order to successfully find the antiderivative of a product.

5. Can I use a calculator to find the antiderivative of a product?

Yes, many calculators have the capability to find the antiderivative of a product using integration techniques. However, it is important to understand the concepts and rules behind finding antiderivatives in order to use a calculator effectively and verify the accuracy of the result.

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