How Do You Integrate x^5 cos(x^3) with Substitution?

  • Thread starter mckallin
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In summary, the conversation is about someone asking for help with solving a specific integration problem involving x^5*cos(x^3). Various suggestions are made, including using integration by parts and making a substitution. The final suggestion is to make the substitution u=x^3 and then integrate by parts, which simplifies the integral and makes it easier to solve.
  • #1
mckallin
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Please Help! Integration!

can anyone help me solve the following integration? thanks a lot.

[tex]\int x^5 cos(x^3) dx[/tex]
 
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  • #2
try to solve it by parts
 
  • #3
I have tried it, but it doesn't work, at least, for me.

If I make [tex] u=cos(x^3), dv=x^5 dx [/tex], the grade of x, which is in the [tex] cos(x^3) [/tex], won't be reduce.

If I make [tex] u=x^5, dv=cos(x^3) dx [/tex], I can't solve the [tex] \int cos(x^3) dx[/tex].

Could you give me some more advice? thanks
 
  • #4
What would you like to have inside the integral in order to have

[tex] \int cos(x^3) dx[/tex]

be solvable? Can you choose slightly different u and dv to accomplish that?
 
  • #5
From [itex](\sin x^3)'=3\,x^2\,\cos x^3[/itex] deduce that [itex]\cos x^3=\frac{1}{3\,x^2}\,(\sin x^3)'[/itex] and use that to integrate by parts.
 
  • #6
Oh, for goodness sake! Make the substitution u= x3, then integrate by parts!

(I notice now that that is essentially what Rainbow Child said.)
 
  • #7
Rainbow Child said:
From [itex](\sin x^3)'=3\,x^2\,\cos x^3[/itex] deduce that [itex]\cos x^3=\frac{1}{3\,x^2}\,(\sin x^3)'[/itex] and use that to integrate by parts.
How it can help?

HallsofIvy said:
Make the substitution u= x3, then integrate by parts!
How this can help?
 
  • #8
fermio said:
How this can help?
I don't think there's any harm in showing the substitution,

[tex]u = x^3 \Rightarrow \frac{du}{dx} = 3x^2 \Rightarrow dx = \frac{du}{3x^2}[/tex]

Hence when we make the substitution the integral becomes,

[tex]\int \frac{x^5}{3x^2}\cos(u)du = \frac{1}{3}\int u\cos(u)du[/tex]

Which is a simple integral to solve.
 
Last edited:

Related to How Do You Integrate x^5 cos(x^3) with Substitution?

1. What is the general process for solving an integration problem?

The general process for solving an integration problem involves using techniques such as substitution, integration by parts, or partial fractions to rewrite the integral in a more manageable form. Then, the integral can be evaluated using known integration formulas or by applying integration rules.

2. How do I know which technique to use for solving this specific integral?

The technique used for solving an integral depends on the form of the integrand. For example, substitution is useful for integrands that involve a composition of functions, while integration by parts is helpful for integrands that involve a product of functions. Familiarity with integration techniques and practice can help you determine the most appropriate approach for a given integral.

3. How do I handle integrals that involve trigonometric functions?

Integrals involving trigonometric functions can often be solved using trigonometric identities and substitution. It is important to recognize common trigonometric patterns and know how to apply them to simplify the integral.

4. What are some tips for successfully solving integration problems?

Some tips for solving integration problems include carefully analyzing the integrand, looking for patterns or opportunities for simplification, and being familiar with common integration formulas. It is also helpful to practice regularly and seek help or guidance when needed.

5. Can I use a calculator to solve this integral?

While some calculators may have built-in integration capabilities, it is important to understand the steps involved in solving an integral by hand. Additionally, calculators may not always provide the most accurate or simplified solution, so it is recommended to use them as a tool for checking your work rather than relying on them to solve the integral entirely.

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