Integration Game (Exponential)

Integration Game (Exponential)
The exponential rule of integration is used for functions where the variable is in the exponent. It’s often broken into two cases, depending on whether the base is the natural constant \(e\) or a general constant \(a\).
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 Exponential. Select one of the four possible answers. 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.
 

 

How to Play the Integration Game (Exponential)

1. Launch the Game: You’ll start at the “Exponential Integration Station” briefing screen.
   
2. Solve Integrals:
   Natural Exponential: Remember \(\int e^{kx} dx = \frac{1}{k}e^{kx}\). Don’t forget to divide by the constant.
   General Exponential: Remember \(\int a^x dx = \frac{a^x}{\ln(a)}\). Divide by the natural log of the base.
   Logarithmic form: Remember \(\int \frac{1}{x} dx = \ln|x|\).
   
3. Instant Feedback:
   Correct: Turns Green + Spacey Chime sound.
   Incorrect: Turns Red + Low Buzz sound. The correct answer will highlight Green so you can learn from the mistake.
   
4. End Game: Abort mission and go back to Menu.
    
   

The Exponential Rule of Integration
The Exponential Rule of Integration is used to find the antiderivative of functions in the form of \(e^x\) or \(a^x\).
Case 1: Integration of the Natural Exponential Function (\(e^x\))
The simplest form of the exponential rule involves the base \(e\) (Euler’s number, approximately \(2.718\)).

The Rule
The integral of \(e^x\) is \(e^x\), plus the constant of integration \(C\).
\(\int e^x dx = e^x + C\)

Why this works: The derivative of \(e^x\) is \(e^x\). Since integration is the reverse of differentiation (finding the antiderivative), the integral of \(e^x\) must also be \(e^x\).

Example
If you need to integrate a simple function involving \(e^x\) and other terms:
\(\int (e^x + 5x^2) dx\)

Applying the rules of integration (linearity and the power rule for the second term):
\(\int e^x dx + \int 5x^2 dx\)
\(e^x + 5 \left(\frac{x^{2+1}}{2+1}\right) + C\)
\(e^x + \frac{5}{3}x^3 + C\)
 

Case 2: Integration of the General Exponential Function (\(a^x\))
This rule applies when the base is any positive constant \(a\) (where \(a > 0\) and \(a \neq 1\)).
The Rule
The integral of \(a^x\) is \(a^x\) divided by the natural logarithm of the base, \(\ln(a)\), plus the constant of integration \(C\).
\(\int a^x dx = \frac{a^x}{\ln(a)} + C\)\)

Why this works: The derivative of \(a^x\) is \(a^x \ln(a)\). To reverse this process, we must divide by the constant factor \(\ln(a)\) during integration.

Example
Let’s find the integral of \(3^x\):
\(\int 3^x dx\)
Here, the base \(a = 3\). Applying the rule:
\(\frac{3^x}{\ln(3)} + C\)

This video gives a clear, step-by-step approach to learn how to use the integration formulas for exponential functions.

• 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 Exponential.

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 Exponential, 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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