Question

find the indefinite integral and check the result by differentiating.

$$\\int e^{3 x}\\left(3+e^{3 x}\ ight)^{2} d x$$

determine an appropriate substitution to simplify the integrand. choose the correct answer below.

a.

$$\\left(3+e^{3 x}\ ight)^{2}$$

b. $$e^{3 x}$$
c. $$3+e^{3 x}$$
d.

$$e^{3 x}\\left(3+e^{3 x}\ ight)^{2}$$

Explanation:

Step1: Analyze substitution options

Let \(u = 3+e^{3x}\). Then \(du=3e^{3x}dx\), and \(e^{3x}dx=\frac{1}{3}du\). The integrand \(e^{3x}(3 + e^{3x})^2dx\) can be rewritten in terms of \(u\).

Step2: Check other options

• Option A: If \(u=(3 + e^{3x})^2\), then \(du = 2(3 + e^{3x})\times3e^{3x}dx\), which is more complex.
• Option B: If \(u = e^{3x}\), then \(du=3e^{3x}dx\), but it doesn't simplify \((3 + e^{3x})^2\) well.
• Option D: \(e^{3x}(3 + e^{3x})^2\) is the integrand itself, not a substitution variable.

Answer:

C. \(3 + e^{3x}\)