Evaluate the Integral Using Trigonometric Substitution

Evaluate the integral $\displaystyle \int x^3 \sqrt{1-x^2} \, dx$ using trigonometric substitution.

Trigonometric substitution is a powerful technique to solve integrals involving square roots of quadratic expressions. This method leverages the Pythagorean identities to transform the integrals into a simpler trigonometric form, which can be easier to evaluate. In this particular problem, the presence of the square root of a quadratic expression, such as one minus x squared, suggests the use of a sine or cosine substitution. By allowing x to equal sine or cosine of theta, the integrand transforms, often simplifying the integration process.

Next, leveraging the Pythagorean identity, the expression under the square root can be rewritten in terms of a trigonometric function. This change of variable not only simplifies the integrand but also changes the differential, dx, into a familiar trigonometric form, making the integral more approachable.

The purpose of using trigonometric substitution lies in reducing complex algebraic integrals to simpler trigonometric forms. After substituting and simplifying, the integral can typically be solved using standard trigonometric integrals. Finally, the solution is completed by converting the results back to the original variable using inverse trigonometric functions. Trigonometric substitution is especially valuable in solving integrals that involve square roots of expressions similar to the Pythagorean triples.

Related Problems