Integration Game (Power Rule)

Integration Game (Power Rule)
The Power Rule is a core technique used in integral calculus to find the antiderivative (the indefinite integral) of functions that are in the form of a variable raised to a constant power.
\(\int x^n dx = \frac{x^{n+1}}{n+1} + C\)
Scroll down the page for a more detailed explanation.
 
In this game, you will need to find the indefinite integral of a given function using the Power Rule. Select one of the four possible answers. The correct answer will be highlighted in green. It includes a scoring system, and clear feedback to help you master this skill.
 

 

How to Play the Integration Game (Power Rule)
This game focuses on using the Power Rule for Integration.
Here’s how to play:

1. Start: Click “Start Practicing”.
   
2. Look at the Problem: You will be given an indefinite integral.
   
3. Select Your Answer: Select the correct integral.
   
4. Check Your Work: If you are right, the answer will be highlighted in green. If you are wrong, the wrong answer will be highlighted in red and correct answer will be highlighted in green.
   
5. Get a New Problem: Click “Next Integral” for a new problem.
   Your score is tracked, showing how many you’ve gotten right.
   
6. Finish Game Check your score. Click “Exit to Menu” to stop the game.
    
   

The Power Rule of Integration
The Power Rule is a core technique used in integral calculus to find the antiderivative (the indefinite integral) of functions that are in the form of a variable raised to a constant power.

The Formula
The Power Rule for Integration states that for any real number \(n\) (except for \(n = -1\)), the integral of \(x\) raised to the power of \(n\) is given by:
\(\int x^n dx = \frac{x^{n+1}}{n+1} + C\)
where:
\(x^n\) is the function being integrated.
\(dx\) indicates that the integration is with respect to the variable \(x\).
\(\frac{x^{n+1}}{n+1}\) is the antiderivative of \(x^n\).
\(C\) is the constant of integration (required for indefinite integrals, as the derivative of any constant is zero).

The Exception: \(n \neq -1\)
It is crucial to note that this rule does not apply when \(n = -1\). If \(n = -1\), the function is \(\frac{1}{x}\), and its integral is handled by a separate rule involving the natural logarithm:
\(\int \frac{1}{x} dx = \int x^{-1} dx = \ln|x| + C\)

How to Use the Power Rule
The Power Rule is used whenever the term you are integrating is a power of \(x\) (or any variable).
Step-by-Step Application
Consider the integral:
\(\int (3x^4 + 2x^2 - 5x + 7) dx\)
Step 1: Use the Linearity of Integration
Break the integral into separate, simpler integrals (this also allows constants to be factored out).
\(\int 3x^4 dx + \int 2x^2 dx - \int 5x dx + \int 7 dx\)
\(3 \int x^4 dx + 2 \int x^2 dx - 5 \int x^1 dx + 7 \int x^0 dx\)
Step 2: Apply the Power Rule to Each Term
For each term \(\int x^n dx\), add 1 to the exponent (\(n+1\)) and divide the entire term by the new exponent (\(n+1\)).
Step 3: Combine the Results and Add the Constant \(C\)
Now, substitute the results back into the equation, remembering to multiply by the constants factored out in Step 1:
\(3 \left(\frac{x^5}{5}\right) + 2 \left(\frac{x^3}{3}\right) - 5 \left(\frac{x^2}{2}\right) + 7 (x) + C\)
Step 4: Simplify the Final Answer
\(\frac{3}{5}x^5 + \frac{2}{3}x^3 - \frac{5}{2}x^2 + 7x + C\)

Handling Roots and Fractions
The Power Rule can be applied to roots and fractions by rewriting them using negative and fractional exponents:
Roots: A root is a fractional exponent.
Example: \(\int \sqrt{x} dx = \int x^{1/2} dx\)
Application: \(\frac{x^{1/2 + 1}}{1/2 + 1} + C = \frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C\)
Fractions (Denominator): A variable in the denominator is a negative exponent.
Example: \(\int \frac{1}{x^3} dx = \int x^{-3} dx\)
Application: \(\frac{x^{-3 + 1}}{-3 + 1} + C = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C\)
Note: Ensure the power is not \(-1\) before applying this rule.
 

2. The Sum and Difference Rules (for combining terms)
The Sum and Difference Rules state that when differentiating a function made up of multiple terms added or subtracted together, you can simply differentiate each term separately and then add or subtract the results.
The Rule:
If \(f(x) = g(x) \pm h(x)\), then the derivative \(f’(x)\) is:
\(f’(x) = g’(x) \pm h’(x)\)
This rule is what allows you to differentiate entire polynomials, such as \(f(x) = 3x^2 + 5x - 8\)
Differentiate \(3x^2\), then \(5x\), and then \(-8\), and combine the results.
 

This video gives a clear, step-by-step approach to learn how to the Integral Power Rule.

• Show Video

 

Derivative Challenge (Polynomials)
Find the derivatives of polynomials using the Power, Sum, Difference, and Constant Multiple rules.

Derivative Challenge (Binomials)
Find the derivatives of the product two binomial functions. Either use the Product Rule or multiply the two binomials first to get a single polynomial and use the Power Rule.

Derivative Challenge (Product Rule)
Find the derivatives of the product a polynomial and a trig function. Use the Product Rule.

Derivative Challenge (Quotient Rule)
Find the derivatives of a function using the Quotient Rule.

Derivative Challenge (Chain Rule)
Find the derivatives of a function using the Chain Rule.

Derivative Rules
Find the derivatives of a function using the Power Rule, Constant Rule, Sum/Difference Rule, Product Rule, Quotient Rule, Chain Rule, and the rules for trigonometric and exponential functions.

Derivative Challenge (Trig Function)
Find the derivatives of Trigonometric Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

Derivative Challenge (Exp, Log)
Find the derivatives of Exponential and Logarithmic Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

Derivative Challenge (Hyperbolic)
Find the derivatives of Hyperbolic Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

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