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

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

∻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.⚹◊What is the integral of $e^{4x}(5+e^{4x})^{8} dx$?⍭ Generate me a similar question◊

Question

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

Accepted Answer

∻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.⚹◊What is the integral of $e^{4x}(5+e^{4x})^{8} dx$?⍭ Generate me a similar question◊

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