Question

find the indefinite integral and check the result by differentiating. determine an appropriate substitution to simplify the integrand. choose the correct answer below. a. $6 + e^{3x}$ b. $e^{3x}(6 + e^{3x})^{4}$ c. $(6 + e^{3x})^{4}$ d. $e^{3x}$ find the indefinite integral. $int e^{3x}(6 + e^{3x})^{4}dx=square$

Explanation:

Step1: Substitution

Let \(u = 6 + e^{3x}\). Then, \(du=3e^{3x}dx\), and \(e^{3x}dx=\frac{1}{3}du\).

Step2: Integral transformation

The integral \(\int e^{3x}(6 + e^{3x})^{4}dx\) becomes \(\frac{1}{3}\int u^{4}du\).
Using the power - rule for integration \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), we have \(\frac{1}{3}\times\frac{u^{5}}{5}+C=\frac{u^{5}}{15}+C\).

Step3: Back - substitution

Substitute \(u = 6 + e^{3x}\) back into the expression. So, \(\int e^{3x}(6 + e^{3x})^{4}dx=\frac{(6 + e^{3x})^{5}}{15}+C\).

Step4: Differentiation check

Differentiate \(y=\frac{(6 + e^{3x})^{5}}{15}+C\) using the chain - rule.
Let \(t = 6+e^{3x}\), then \(y=\frac{t^{5}}{15}+C\). \(\frac{dy}{dt}=\frac{5t^{4}}{15}=\frac{t^{4}}{3}\), and \(\frac{dt}{dx}=3e^{3x}\).
By the chain - rule \(\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}\), \(\frac{dy}{dx}=\frac{(6 + e^{3x})^{4}}{3}\cdot3e^{3x}=e^{3x}(6 + e^{3x})^{4}\).

Answer:

\(\frac{(6 + e^{3x})^{5}}{15}+C\)