https://file.aiquickdraw.com/mathbot/file/1726704762568l7rptxdb.jpg

The integral provided in the image is:

\[
\int \left( x + a + t^2 \right) dx
\]

where \(a\) and \(t\) are constants.

To solve this indefinite integral, we can split the terms inside the integrand and integrate each one separately:

\[
\int \left( x + a + t^2 \right) dx = \int x \, dx + \int a \, dx + \int t^2 \, dx
\]

1. \(\int x \, dx = \frac{x^2}{2}\)
2. \(\int a \, dx = a \cdot x\), since \(a\) is a constant
3. \(\int t^2 \, dx = t^2 \cdot x\), since \(t^2\) is also a constant (independent of \(x\))

Now, summing these up:

\[
\int \left( x + a + t^2 \right) dx = \frac{x^2}{2} + ax + t^2x + C
\]

where \(C\) is the constant of integration.

So the solution is:

\[
\frac{x^2}{2} + ax + t^2x + C
\]

Do you need further details or have any questions?

Here are some related questions you might want to explore:

1. How do you handle the integration of products of functions?
2. What is the difference between indefinite and definite integrals?
3. How do constants affect the integration process?
4. Can you explain when to use integration by substitution?
5. 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.

Find the indefinite integral \( \int (x + a + t^2) \, dx \), where \(a\) and \(t\) are constants.

Learn how to find the indefinite integral \( \int (x + a + t^2) \, dx \) with a step-by-step solution. This problem covers key calculus concepts, including integration of linear combinations and handling constants during integration. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation