```
Evaluate the indefinite integral. (Remember the constant of integration.)
$\int e^{4 x}\left(5+e^{4 x}\right)^{8} d x=$
$\square$
```

Sign in to unlock the answer

Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To integrate the function $e^{4x}(5+e^{4x})^8$, we can use substitution. Let $u = 5+e^{4x}$, then $du = 4e^{4x}dx$

step 2

To adjust for the $4$ in $du$, we multiply and divide the integral by $4$, giving us $\frac{1}{4} \int u^8 du$

step 3

Now we integrate $u^8$ with respect to $u$ to get $\frac{1}{4} \cdot \frac{u^9}{9} + C$, where $C$ is the constant of integration

step 4

Substituting back for $u$, we get $\frac{1}{36} e^{4x}(5+e^{4x})^9 + C$

Answer

$\frac{1}{36} e^{4x}(5+e^{4x})^9 + C$

Key Concept

Integration by Substitution

Explanation

To integrate a composite function like $e^{4x}(5+e^{4x})^8$, we use substitution to simplify the integral into a form that is easier to integrate directly.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question