Indefinite integral substitution

In summary, substitution is used in indefinite integrals to simplify the integration process by replacing a complex expression with a simpler one. The choice of substitution variable is typically based on the original expression and common choices include trigonometric functions, logarithms, and exponential functions. There are two main types of substitution, u-substitution and trigonometric substitution. However, substitution may not always be applicable or lead to a simpler solution, so it is important to carefully consider the integral before using it. Common mistakes to avoid when using substitution include not adjusting the limits of integration properly and not checking the final solution for equivalence to the original integral.
  • #1
sapiental
118
0
evaluate the indefinite integral ((e^x)/((e^x)+1))dx

I let u = ((e^x)+1)

then

du = (e^x)dx

which occurs in the original equation so..

indefinite ingegral ((u^-1)du)

taking the antiderivative I get 1 + C

is this right? thanks!
 
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  • #2
[tex] \int \frac{e^{x}}{e^{x}+1} dx [/tex]

it should be [tex] \ln|e^{x}+1| + C [/tex] because it is of the form [tex] \int \frac{du}{u}[/tex]
 
Last edited:

Related to Indefinite integral substitution

1. What is the purpose of using substitution in indefinite integrals?

Substitution is used to simplify integrals by replacing a complex expression with a simpler one. This allows for easier integration and often leads to a more straightforward solution.

2. How do you choose the substitution variable in an indefinite integral?

The substitution variable is typically chosen to be a part of the original expression that can be easily integrated. Common choices include trigonometric functions, logarithms, and exponential functions.

3. What are the different types of substitution used in indefinite integrals?

The two main types are u-substitution and trigonometric substitution. U-substitution involves replacing a variable with a single term, while trigonometric substitution involves replacing a variable with a trigonometric function.

4. Can substitution be used in all indefinite integrals?

No, substitution may not always be applicable or may not lead to a simpler solution. It is important to carefully consider the integral before deciding to use substitution.

5. Are there any common mistakes to avoid when using substitution in indefinite integrals?

One common mistake is not properly adjusting the limits of integration after substituting. It is also important to carefully check the final solution to ensure that it is equivalent to the original integral.

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