Substitution and Integration by Parts

In summary, the conversation discusses solving the integral ∫x^{7}cos(x^{4})dx by first making a substitution and then using integration by parts. The equations for substitution and integration by parts are also mentioned. The person attempted to solve the integral but got stuck and asked for help. They later realized their mistake and thanked the person for their assistance.
  • #1
sashab
12
0

Homework Statement


First make a substitution and then use integration by parts to evaluate the integral.

∫x[itex]^{7}[/itex]cos(x[itex]^{4}[/itex])dx

Homework Equations



Equation for Substitution: ∫f(g(x))g'(x)dx = ∫f(u)du
Equation for Integration by Parts: ∫udv = uv - ∫vdu

The Attempt at a Solution



So here's my attempted solution
tumblr_n1aepoItwY1tsd2vco1_500.jpg


I made a substitution and tried using integration by parts twice but I got stuck on the last line since it turns out to be zero... I know I went wrong somewhere but I can't seem to find my mistake. Any help would be really appreciated! Thanks :)
 
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  • #2
The second line is incorrect - when you made the u substitution you did not use your expression for dx in terms of du.
 
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  • #3
Start by writing the integral as [itex]\int x^4cos(x^4)(x^3dx)[/itex] and it is clearer.
 
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  • #4
I see my mistake now! Thanks for the help :)
 

Related to Substitution and Integration by Parts

1. What is the difference between substitution and integration by parts?

Substitution and integration by parts are both techniques used in solving integrals, but they differ in their approach. Substitution involves replacing a variable in the integral with a new variable, while integration by parts involves breaking down the integral into two parts and integrating each part separately.

2. When should I use substitution and when should I use integration by parts?

Substitution is typically used when the integral contains a single variable or a single function, while integration by parts is used when the integral contains a product of two functions.

3. How do I know which substitution to use?

In order to choose the correct substitution, you must first identify the "inside" function in the integral. This is the function that is being raised to a power or being multiplied by another function. The substitution should then be chosen so that the derivative of the new variable matches the "inside" function.

4. What is the formula for integration by parts?

The formula for integration by parts is ∫u dv = uv - ∫v du. In this formula, u and v represent the two parts of the integral, with u being the "inside" function and v being the other function. The du and dv represent the derivatives of u and v, respectively.

5. Can I use both substitution and integration by parts in the same integral?

Yes, it is possible to use both techniques in the same integral. This is known as the "double substitution" or "substitution by parts" method. It involves using substitution to simplify the integral, and then using integration by parts to solve the remaining integral.

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