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:

(x+a+t2)dx\int \left( x + a + t^2 \right) dx

where aa and tt are constants.

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

(x+a+t2)dx=xdx+adx+t2dx\int \left( x + a + t^2 \right) dx = \int x \, dx + \int a \, dx + \int t^2 \, dx

  1. xdx=x22\int x \, dx = \frac{x^2}{2}
  2. adx=ax\int a \, dx = a \cdot x, since aa is a constant
  3. t2dx=t2x\int t^2 \, dx = t^2 \cdot x, since t2t^2 is also a constant (independent of xx)

Now, summing these up:

(x+a+t2)dx=x22+ax+t2x+C\int \left( x + a + t^2 \right) dx = \frac{x^2}{2} + ax + t^2x + C

where CC is the constant of integration.

So the solution is:

x22+ax+t2x+C\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.

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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