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Evaluate the integral ∫ e^(3x) dx.

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Question: Evaluate the integral ∫ e^(3x) dx.

Options:

  1. (1/3)e^(3x) + C
  2. (1/3)e^(3x)
  3. 3e^(3x) + C
  4. e^(3x) + C

Correct Answer: (1/3)e^(3x) + C

Solution: 

The integral of e^(kx) is (1/k)e^(kx). Here, k = 3, so ∫ e^(3x) dx = (1/3)e^(3x) + C.

Evaluate the integral ∫ e^(3x) dx.

Practice Questions

Q1
Evaluate the integral ∫ e^(3x) dx.
  1. (1/3)e^(3x) + C
  2. (1/3)e^(3x)
  3. 3e^(3x) + C
  4. e^(3x) + C

Questions & Step-by-Step Solutions

Evaluate the integral ∫ e^(3x) dx.
  • Step 1: Identify the integral you need to evaluate, which is ∫ e^(3x) dx.
  • Step 2: Recognize that the integral of e^(kx) is (1/k)e^(kx), where k is a constant.
  • Step 3: In this case, k is 3 because we have e^(3x).
  • Step 4: Substitute k = 3 into the formula: (1/3)e^(3x).
  • Step 5: Don't forget to add the constant of integration, C, to your answer.
  • Step 6: Write the final answer: ∫ e^(3x) dx = (1/3)e^(3x) + C.
  • Integration of Exponential Functions – The integral of an exponential function of the form e^(kx) is (1/k)e^(kx) + C, where k is a constant.
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