Integral Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Integral.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The reverse operation of differentiation; it also computes the exact area under a curve between two points.

If derivative gives rate, integral gives total. Derivative of position = velocity; integral of velocity = position.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Integration accumulates; differentiation rates. They're inverses.

Common stuck point: Always write +C for indefinite integrals—omitting it loses the entire family of antiderivatives.

Sense of Study hint: Ask yourself: what function, when differentiated, gives me this integrand? Check by differentiating your answer.

Worked Examples

Example 1

easy
Find \int (4x^3 + 6x) \, dx

Solution

  1. 1
    Apply the power rule for integration: \int x^n \, dx = \frac{x^{n+1}}{n+1} + C.
  2. 2
    For 4x^3: \frac{4x^4}{4} = x^4.
  3. 3
    For 6x: \frac{6x^2}{2} = 3x^2.
  4. 4
    Combine with the constant of integration: x^4 + 3x^2 + C.

Answer

x^4 + 3x^2 + C
Integration reverses differentiation. The power rule for integration adds 1 to the exponent and divides by the new exponent. Always include the constant C for indefinite integrals.

Example 2

medium
Evaluate \int_0^2 (3x^2 + 1) \, dx

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find \int (5x^2 - 3x + 7) \, dx

Example 2

hard
Find \int \frac{1}{x} \, dx and explain why the standard power rule does not apply.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

derivative