Math Problem Statement
Find the indefinite integral \( \int (x + a + t^2) \, dx \), where \(a\) and \(t\) are constants.
Solution
The integral provided in the image is:
where and are constants.
To solve this indefinite integral, we can split the terms inside the integrand and integrate each one separately:
- , since is a constant
- , since is also a constant (independent of )
Now, summing these up:
where is the constant of integration.
So the solution is:
Do you need further details or have any questions?
Here are some related questions you might want to explore:
- How do you handle the integration of products of functions?
- What is the difference between indefinite and definite integrals?
- How do constants affect the integration process?
- Can you explain when to use integration by substitution?
- What is the geometric interpretation of an indefinite integral?
Tip: When constants appear in an integral, treat them like any other fixed number during the integration process—they remain unaffected by the variable of integration.
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Math Problem Analysis
Mathematical Concepts
Calculus
Integration
Formulas
\( \int x \, dx = \frac{x^2}{2} \)
\( \int a \, dx = ax \)
\( \int t^2 \, dx = t^2x \)
Theorems
Linear Combination of Integrals
Suitable Grade Level
Grades 11-12