Math Problem Statement
Find the indefinite integral.
∫(e^(3x)−x)^2dx
Answer:
Solution
To solve the indefinite integral
we begin by expanding the integrand.
Step 1: Expand the expression
Step 2: Break down the integral into simpler parts
Now we can rewrite the integral as the sum of three integrals:
Step 3: Solve each integral
-
Integral of :
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Integral of : To solve , we use integration by parts. Let: Then, and . Now, apply the integration by parts formula :
The remaining integral is:
Thus:
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Integral of :
Step 4: Combine all results
Now, combine the results of all three integrals:
Simplify the second term:
Thus, the final result is:
where is the constant of integration.
Would you like further clarification on any step or method? Here are a few related questions:
- What is integration by parts, and how does it work?
- How do we handle integrals of products like ?
- What is the general approach to expanding expressions before integrating?
- How do you determine when to use substitution versus integration by parts?
- Can the result be simplified further or expressed in a more compact form?
Tip: When expanding complex terms before integration, always check if any of the parts can be simplified or factored to make the integration process easier.
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Math Problem Analysis
Mathematical Concepts
Integration
Expansion
Integration by Parts
Exponential Functions
Formulas
∫e^(ax)dx = e^(ax)/a
Integration by parts: ∫udv = uv - ∫vdu
Theorems
Integration by parts
Suitable Grade Level
Grades 11-12