A Comprehensive Guide: Integrating x^2 5^5 Using Substitution and Binomial Theorem

Introduction

When dealing with integrals involving polynomials and specific constants, such as the function ( x^2 - 5^5 ), it is essential to understand various integration techniques to arrive at a solution. This guide will walk you through the process of integrating the function ( x^2 - 5^5 ) using both substitution and the binomial theorem. By following these steps, you'll be able to solve similar problems efficiently.

Step-by-Step Integration Using Substitution

1. Substitution

To integrate ( x^2 - 5^5 ), we can use the method of substitution. Let's start by setting ( u x^2 - 5 ).

1.1 Derivative of ( u )

The derivative of ( u ) with respect to ( x ) is:

[ frac{du}{dx} 2x ]

This can be rearranged to express ( dx ) in terms of ( du ):

[ du 2x , dx quad Rightarrow quad dx frac{du}{2x} ]

1.2 Rewriting the Integral

Given that ( x^2 u - 5 ), we can substitute ( x ) in terms of ( u ):

[ int x^2 5^5 , dx int (u^5) cdot frac{du}{2sqrt{u - 5}} ]

This integral now involves ( u^5 ) and a term involving ( sqrt{u - 5} ).

1.3 Integrate the Simplified Expression

Although this form is simpler, it may require further integration techniques. For an indefinite integral, we can go back to the original expression and use the binomial theorem.

Integrating Using the Binomial Theorem

Another approach to integration is to expand the polynomial ( x^2 - 5^5 ) using the binomial theorem. This is particularly useful because it can simplify the integration process significantly.

2. Applying the Binomial Theorem

The binomial theorem states that:

[ (a b)^n sum_{k0}^{n} binom{n}{k} a^{n-k} b^k ]

For ( (x^2 - 5^5) ), we can write:

[ x^2 - 5^5 sum_{k0}^{5} binom{5}{k} (x^2)^k (-5)^{5-k} ]

Expanding this, we get:

[ x^2 - 5^5 sum_{k0}^{5} binom{5}{k} 5^{5-k} x^{2k} ]

To integrate each term, we use the standard integration rule:

[ int x^ndx frac{x^{n 1}}{n 1} C ]

Thus, for each term in the series:

[ int 5^{5-k} x^{2k} , dx 5^{5-k} cdot frac{x^{2k 1}}{2k 1} C ]

Summing these results, we obtain:

[ int x^2 5^5 , dx sum_{k0}^{5} binom{5}{k} 5^{5-k} cdot frac{x^{2k 1}}{2k 1} C ]

This is the final result of the indefinite integral, where ( C ) is the constant of integration.

3. Numerical Evaluation

For a numerical evaluation, we can use the steps above to compute specific values of the integral. Alternatively, if you are aware of the binomial theorem, you can use it to expand and integrate the polynomial term-by-term.

Conclusion

Integrating the function ( x^2 - 5^5 ) can be approached through substitution, binomial theorem, or direct polynomial integration. By mastering these techniques, you will be well-equipped to handle similar integration problems. Remember that understanding the binomial theorem and substitution methods are key to solving these types of integrals effectively.