Substitution Rule for Indefinite Integrals

In summary, to find the antiderivative of e^-^7^x, we use the formula -2/7 * e^-^7^x + C. However, it is important to note that there is no need to divide by another -7 as it is already in standard form.
  • #1
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Homework Statement


[tex]\int2e^-^7^xdx[/tex]


Homework Equations


None


The Attempt at a Solution



[tex](\frac{-2}{7})(\frac{e^-^7^x}{-7})+C[/tex]


This is as far as I can go, but the answer is:

[tex]\frac{-2e^-^7^x}{7}+C[/tex]
 
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  • #2
How did you get two 1/7 terms?
 
  • #3
I took the antiderivative of [tex]e^-^7^x[/tex].
 
  • #4
temaire said:
I took the antiderivative of [tex]e^-^7^x[/tex].
So, why do you have 1/7 * 1/7? Just do it again from scratch and you'll probably see what you did wrong.
 
  • #5
[tex]2\int e^{-7x}dx[/tex]

[tex]-\frac 2 7\int -7e^{-7x}dx[/tex]

You don't need to divide by another -7. It's already in standard form!
 
  • #6
I've got it now, thanks.:smile:
 

What is the Substitution Rule for Indefinite Integrals?

The Substitution Rule for Indefinite Integrals is a method of integration used to simplify integrals by substituting a new variable for the original variable. This allows for the integral to be rewritten in terms of the new variable, making it easier to solve.

How do you use the Substitution Rule for Indefinite Integrals?

To use the Substitution Rule, you must first identify the inner function and its derivative in the integral. Then, choose a new variable to substitute for the inner function, and replace all instances of the inner function with the new variable. Finally, rewrite the integral in terms of the new variable and solve.

Why is the Substitution Rule useful in integration?

The Substitution Rule is useful because it allows for the integration of more complicated functions by simplifying them into a form that is easier to solve. It also helps to avoid the use of complicated integration techniques such as integration by parts.

What are the common mistakes made when using the Substitution Rule?

One common mistake is incorrect substitution, where the new variable is not chosen correctly or the original variable is not replaced correctly. Another mistake is forgetting to include the derivative of the inner function in the integral. It is also important to remember to change the limits of integration when using the Substitution Rule.

Can the Substitution Rule be used for definite integrals?

Yes, the Substitution Rule can also be used for definite integrals. However, the limits of integration must also be changed to match the new variable. This can be done by substituting the original limits into the new variable expression.

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