Integrating ( frac{x^3}{sqrt{81-x^2}} ) without Substitution: A Direct Approach

Integrating complex functions without resorting to traditional methods such as substitution can be a challenging but satisfying exercise. This article will guide you through the process of integrating ( frac{x^3}{sqrt{81-x^2}} ) directly using integration by parts. This method avoids the complication of substitution, demonstrating a cleaner and more straightforward approach to solving such integrals.

Step-by-Step Integration by Parts

The given integral is:

( int frac{x^3}{sqrt{81-x^2}} dx )

Let's use integration by parts, where:

( u x^2 ) and ( dv frac{x}{sqrt{81-x^2}} dx )

Choosing ( u ) and ( dv )

For this problem, we set ( u x^2 ), which implies ( du 2x dx ). The choice of ( dv frac{x}{sqrt{81-x^2}} dx ) is the crucial step. To integrate ( dv ), we use a direct approach without substitution:

[ v int frac{x}{sqrt{81-x^2}} dx ]

Consider the integral of ( frac{x}{sqrt{81-x^2}} dx ). We can use integration by substitution, but since the problem specifically asks to avoid substitution, we can use a known integral:

[ int frac{x}{sqrt{81-x^2}} dx -sqrt{81-x^2} C ]

Therefore:

[ v -sqrt{81-x^2} ]

Applying Integration by Parts Formula

The integration by parts formula is:

[ int u dv uv - int v du ]

Substituting ( u x^2 ), ( du 2x dx ), and ( v -sqrt{81-x^2} ) into the formula, we get:

[ int frac{x^3}{sqrt{81-x^2}} dx -x^2 sqrt{81-x^2} - int -sqrt{81-x^2} cdot 2x dx ]

Which simplifies to:

[ int frac{x^3}{sqrt{81-x^2}} dx -x^2 sqrt{81-x^2} 2 int x sqrt{81-x^2} dx ]

To integrate ( 2 int x sqrt{81-x^2} dx ), let's use a simpler approach without substitution:

[ 2 int x sqrt{81-x^2} dx -frac{2}{3}(81-x^2)^{frac{3}{2}} C ]

This is obtained by recognizing the integral as a form of ( int -2x sqrt{81-x^2} dx ), which integrates to ( -frac{(81-x^2)^{frac{3}{2}}}{3} ).

Therefore, combining all parts, the final integration is:

[ int frac{x^3}{sqrt{81-x^2}} dx -x^2 sqrt{81-x^2} - frac{2}{3}(81-x^2)^{frac{3}{2}} C ]

Simplifying further:

[ int frac{x^3}{sqrt{81-x^2}} dx -frac{1}{3} sqrt{81-x^2} (3x^2 2(81-x^2)) C ]

Which simplifies to:

[ int frac{x^3}{sqrt{81-x^2}} dx -frac{1}{3} sqrt{81-x^2} left(3x^2 - 2x^2 frac{162}{x^2}right) C ]

This can be further simplified to:

[ int frac{x^3}{sqrt{81-x^2}} dx -frac{1}{3} sqrt{81-x^2} left(x^2 frac{162}{x^2}right) C ]

Conclusion

This method provides a direct and clear path to solving the given integral without resorting to substitution. It demonstrates the power of integration by parts in tackling complex integrals.

For verification, the final result can be differentiated to ensure it correctly reproduces the original function.

By mastering such techniques, you can enhance your problem-solving skills in calculus and integration.