# How do I integrate (e^3lnx + e^3x)dx without a calculator?

• boardinchic
In summary, to integrate (e^3lnx + e^3x)dx, you can separate it into two integrals of e^(3*lnx) and e^3x, with the latter being relatively straightforward. To solve e^(3*lnx), you can use the property that 3*lnx is equivalent to ln(x^3), and then use the inverse property of ln(y) to e^y.
boardinchic

## Homework Statement

I have to integrate (e^3lnx + e^3x)dx

## The Attempt at a Solution

I have the answer and its (1/4)x^4 + (e^3x)/3 + C.
I don't know how to get to the answer without the help of my calculator.

boardinchic said:
I don't know how to get to the answer without the help of my calculator.

I'm not surprised.

Do you mean e^[3ln(x)]?

Try and simplify that. Remember ln(y) is the inverse operation to e^y

boardinchic said:
I have to integrate (e^3lnx + e^3x)dx

So, is this e^3*lnx or e^(3*lnx)?

First, you can separate them into two integrals of e^(3*lnx) and e^3x. I assume you can solve the second one.

So, for the integral of e^(3*lnx), I'll give you a hint.

3*lnx = ln(x^3)

Thank you. I got that the e^lnx = x, but didn't know what to do with the 3. I get it now.

## 1. What is the process for integrating (e^3lnx + e^3x)dx?

The process for integrating (e^3lnx + e^3x)dx involves using the power rule and substitution to simplify the expression. First, rewrite e^3lnx as x^3 and e^3x as (e^3)^x. Then, use the power rule to integrate x^3, which becomes x^4/4. For (e^3)^x, use substitution by letting u = e^3x and du = 3e^3x dx. This simplifies the expression to (1/3)e^3x. Finally, integrate (1/3)e^3x to get (1/9)e^3x + C. The final answer is (1/9)e^3x + x^4/4 + C.

## 2. Can the integral of (e^3lnx + e^3x)dx be solved using any other method?

Yes, the integral of (e^3lnx + e^3x)dx can also be solved using integration by parts. Let u = e^3x and dv = dx, and use the formula for integration by parts: ∫udv = uv - ∫vdu. This method will also result in the answer (1/9)e^3x + x^4/4 + C.

## 3. What is the purpose of using substitution when integrating (e^3lnx + e^3x)dx?

Substitution is used to simplify the expression and make it easier to integrate. In this case, substituting u = e^3x and using the power rule for integration helps to simplify the expression and make it easier to solve.

## 4. Is there any special technique that can be used to integrate (e^3lnx + e^3x)dx?

Yes, other than substitution and integration by parts, the integral of (e^3lnx + e^3x)dx can also be solved using the technique of partial fractions. This method involves breaking down the fraction into simpler fractions and then integrating each term separately. However, this method may not always be the most efficient or straightforward way to solve the integral.

## 5. What are the limits of integration for the integral of (e^3lnx + e^3x)dx?

The limits of integration will depend on the specific problem or context in which the integral is being used. It is important to identify the limits of integration before solving the integral, and to adjust the final answer accordingly. If no limits are given, the integral is typically solved with indefinite integration and the answer will include the constant of integration, C.

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