Integrating 2^x * e^x and Tips for Solving the Integral

In summary, the conversation discusses how to integrate an expression of the form 2^x * e^x using integration by parts and rewriting it as eu. It also mentions a quote from Laplace about the difficulties of integration.
  • #1
jaykeegan
2
0
I'm wondering how to integrate something of the form 2^x * e^x.


I tried using integration by parts, ∫u dv = v*u - ∫v du. The problem is no matter which term I choose to integrate and which to differentiate, the final solution still has an integral with a product of terms in the form (2^x) and (e^x). (∫2^x = 2^x / ln2 and (2^x)' = 2^x * ln2).
 
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  • #2
Rewrite as eu, where u is a function of x.
 
  • #3
jaykeegan said:
I'm wondering how to integrate something of the form 2^x * e^x.


I tried using integration by parts, ∫u dv = v*u - ∫v du. The problem is no matter which term I choose to integrate and which to differentiate, the final solution still has an integral with a product of terms in the form (2^x) and (e^x). (∫2^x = 2^x / ln2 and (2^x)' = 2^x * ln2).

##2^x=e^{log(2) x}##.
 
  • #4
Ah okay thanks people. Both of these methods will simplify it. It's been so long since I've had to do calculus, haha. Forgotten all the different ways of looking at an expression.
 
  • #5
I found this quotation from Laplace, which you might like:

"Nature laughs at the difficulties of integration."
 

FAQ: Integrating 2^x * e^x and Tips for Solving the Integral

1. What is the formula for integrating 2^x * e^x?

The formula for integrating 2^x * e^x is ∫2^x * e^x dx = 2^x * e^x / ln(2) + C, where C is the constant of integration.

2. How do you solve the integral of 2^x * e^x?

To solve the integral of 2^x * e^x, you can use integration by parts, where u = 2^x and dv/dx = e^x. This will result in the formula mentioned in the previous answer.

3. Is there an alternative method for integrating 2^x * e^x?

Yes, you can also use substitution by letting u = 2^x. This will result in the integral becoming ∫e^u du, which can be easily solved using the power rule.

4. Can the integral of 2^x * e^x be solved using a calculator?

Yes, most scientific calculators have the capability to solve integrals, including the integral of 2^x * e^x. However, it is still important to understand the concepts and steps involved in solving the integral.

5. What are the applications of integrating 2^x * e^x in science?

Integrating 2^x * e^x can be used in various fields of science, including physics, chemistry, biology, and economics. It can help in understanding growth and decay in natural and artificial systems, as well as predicting future trends.

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