# Easiest way to take the integral of(involving substitution)

• MelissaJL
In summary, the easiest way to take the integral of \int\frac{(6+e^{x})^{2}dx}{e^{x}} is to multiply out (6+e^x)^2 and then integrate the resulting expression, which is ex+12x-36e-x+C.
MelissaJL
What is the easiest way to take the integral of:
$\int$$\frac{(6+e^{x})^{2}dx}{e^{x}}$

I have been having quite some difficulties with this one but here is my work so far:

Let u=e^{x}, du=e^{x}dx
=$\int$$\frac{(u+6)^{2}du}{u^{2}}$
Then let s=u+6 ∴ u=s-6, ds=du
=$\int$$\frac{s^{2}ds}{(s-6)^{2}}$
=$\int$$\frac{s^{2}ds}{36-12s+s^{2}}$

At this point I find myself lost and am not sure what to do. Is there an easier way to solve this integral? Also, if this is the only way to solve it how do I finish it from here? Thank you :)

MelissaJL said:
What is the easiest way to take the integral of:
$\int$$\frac{(6+e^{x})^{2}dx}{e^{x}}$

I have been having quite some difficulties with this one but here is my work so far:

Let u=e^{x}, du=e^{x}dx
=$\int$$\frac{(u+6)^{2}du}{u^{2}}$
Then let s=u+6 ∴ u=s-6, ds=du
=$\int$$\frac{s^{2}ds}{(s-6)^{2}}$
=$\int$$\frac{s^{2}ds}{36-12s+s^{2}}$

At this point I find myself lost and am not sure what to do. Is there an easier way to solve this integral? Also, if this is the only way to solve it how do I finish it from here? Thank you :)

I think the easiest way to do it is just multiply out (6+e^x)^2.

Dick said:
I think the easiest way to do it is just multiply out (6+e^x)^2.

Haha, I don't know why I didn't just do that, apparently I want to just make things more complicated than they really are...
So then multiplying out gives me:
$\frac{(6+e^{x})^{2}}{e^{x}}$ = $\frac{e^{2x}+12e^{x}+36}{e^{x}}$
$\int$ (ex+12+36e-x)dx
= ex+12x-36e-x+C

Thank you.

## 1. What is substitution in integration?

Substitution is a technique used in integration to simplify the integrand (the function being integrated) by replacing a variable with a new one. This new variable is chosen in such a way that it makes the integration process easier.

## 2. How do I choose the right substitution for a given integral?

The key to choosing the right substitution is recognizing patterns in the integrand. Look for expressions that can be rewritten in terms of a new variable, such as trigonometric functions or exponential functions. It also helps to consider the derivative of the new variable and how it can be used to simplify the integrand.

## 3. Can I always use substitution to solve an integral?

No, substitution is not always the best or easiest method for solving an integral. It is important to also consider other techniques such as integration by parts or partial fractions, depending on the complexity of the integrand.

## 4. What is the general process for using substitution in integration?

The general process for using substitution in integration involves the following steps: 1) Identify the appropriate substitution by recognizing patterns in the integrand, 2) Substitute the new variable and its derivative into the integral, 3) Simplify the integrand using the new variable, 4) Integrate the simplified expression, and 5) Substitute the original variable back into the solution.

## 5. Are there any common mistakes to avoid when using substitution in integration?

Yes, some common mistakes to avoid when using substitution in integration include choosing the wrong substitution or forgetting to substitute the new variable in the final solution. It is also important to pay attention to limits of integration and make the appropriate changes when using substitution.

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