Question

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

Explanation:

Step1: Recall substitution rule

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

Step2: Rewrite the integral

The integral \(\int e^{3x}(6 + e^{3x})^{4}dx\) becomes \(\frac{1}{3}\int u^{4}du\).

Step3: Integrate \(u\)

Using the power rule \(\int u^{n}du=\frac{u^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(n = 4\), we have \(\frac{1}{3}\times\frac{u^{5}}{5}+C=\frac{u^{5}}{15}+C\).

Step4: Substitute back \(u\)

Substituting \(u = 6 + e^{3x}\) back, the integral is \(\frac{(6 + e^{3x})^{5}}{15}+C\).

Step5: Differentiate the result

Using the chain rule \((f(g(x)))^\prime=f^\prime(g(x))\cdot g^\prime(x)\). Let \(y=\frac{(6 + e^{3x})^{5}}{15}+C\). Then \(y^\prime=\frac{5(6 + e^{3x})^{4}\times3e^{3x}}{15}=e^{3x}(6 + e^{3x})^{4}\).

Answer:

A. \(6 + e^{3x}\)