Find primitives of function (1+4x)/sqrt(1 + x + 2x^2)

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In summary, a primitive of a function is another function whose derivative is equal to the original function, also known as an antiderivative. To find the primitive of a function, you can use integration techniques such as substitution, integration by parts, or partial fractions. The process for finding the primitive of (1+4x)/sqrt(1 + x + 2x^2) involves rewriting the function and using the substitution method. The primitive of a function can have multiple solutions due to the addition of a constant term. Finding the primitive of a function is significant as it allows us to solve problems involving the original function, such as finding the area under the curve, and is useful in solving differential equations and other areas of mathematics and science
  • #1
Taryn
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This is a practice exam question that I have been given!
Find the primitives of the functions
(1+4x)/(sqrt(1+x+2x^2))

My question is 1. is a primitive the antiderivative? I don't remember my lecturer using primitive during our course!
 
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  • #2
It is basically asking you to find:

[tex] \int \frac{1+4x}{\sqrt{1+x+2x^{2}}} \; dx [/tex]

A primitive is an antiderivative.

So set [tex] u = 1+x+2x^{2} [/tex]

Then [tex] du = 4x+1 \; dx [/tex] and you end up with [tex] \int u^{-\frac{1}{2}} \; du [/tex]. All the primitives mean that you add the integration constant [tex] C [/tex].
 
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  • #3
ahhh thanks!
 

Related to Find primitives of function (1+4x)/sqrt(1 + x + 2x^2)

1. What is a primitive of a function?

A primitive of a function is another function whose derivative is equal to the original function. It is also known as an antiderivative.

2. How do you find the primitive of a function?

To find the primitive of a function, you can use integration techniques such as substitution, integration by parts, or partial fractions. In some cases, you may also need to use special rules or formulas.

3. What is the process for finding the primitive of (1+4x)/sqrt(1 + x + 2x^2)?

To find the primitive of (1+4x)/sqrt(1 + x + 2x^2), you can first rewrite the function as (1+2x+2x^2)/sqrt(1 + x + 2x^2). Then, you can use the substitution method, letting u = 1 + x + 2x^2. After substituting and simplifying, you can then integrate using the power rule and the constant multiple rule.

4. Can the primitive of a function have multiple solutions?

Yes, the primitive of a function can have multiple solutions. This is because adding a constant term to the primitive does not affect its derivative. Therefore, the primitive of a function is not unique.

5. What is the significance of finding the primitive of a function?

Finding the primitive of a function is important because it allows us to solve problems involving the original function, such as finding the area under the curve. It also helps in solving differential equations and in other areas of mathematics and science.

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