Integration of 3x2 / √(1-x2): A Step-by-Step Guide with Trigonometric Substitution

Integrating functions can be a challenging task, especially when dealing with tricky polynomial fractions under a square root or fraction. In such cases, trigonometric substitution can be a powerful tool to simplify the problem. In this article, we will walk through how to integrate the function 3x2 / √(1-x2) using the substitution x sinθ. By transforming the integral into a more manageable form, we can follow through until the answer falls out.

Introduction to the Problem

Consider the integral:

∫(3x2 / √(1-x2)) dx.

This integral looks complicated due to the square root in the denominator, which is a common obstacle when dealing with such functions. To tackle this, we'll utilize a trigonometric identity and substitution. Let's get started.

Step 1: Substitution

Let's introduce a substitution:

x sinθ

This substitution is chosen because it allows us to simplify the square root in the denominator using the Pythagorean identity:

1 - sin2θ cos2θ

With this substitution, we also need to calculate the differential dx in terms of θ:

dx cosθ dθ

Step 2: Rewrite the Integral

We will now rewrite the original integral using the substitution and the differential. By substituting x sinθ and dx cosθ dθ, the integral becomes:

∫(3sin2θ / √(1 - sin2θ)) cosθ dθ

Recall that 1 - sin2θ cos2θ, so the integral simplifies to:

∫(3sin2θ / cosθ) cosθ dθ

The cosθ terms in the numerator and denominator cancel out, leaving us with:

3∫sin2θ dθ

Step 3: Utilize a Known Integral

The integral ∫sin2θ dθ can be solved using the known identity:

sin2θ (1 - cos2θ) / 2

Substituting this identity into the integral, we get:

3∫(1 - cos2θ) / 2 dθ

This simplifies to:

(3/2)∫(1 - cos2θ) dθ

We can split this into two integrals:

(3/2)[∫dθ - ∫cos2θ dθ]

The first integral is:

(3/2)∫dθ (3/2)θ

The second integral requires a substitution u 2θ, so:

∫cos2θ dθ (1/2)sin2θ

Putting it all together:

(3/2)(θ - (1/2)sin2θ)

Remember that sin2θ 2sinθcosθ, so:

(3/2)(θ - sinθcosθ)

Step 4: Convert Back to x

Since x sinθ, we need to express the solution in terms of x. We have:

θ arcsin(x)

The cosine of θ can be found using the right triangle formed by x sinθ opposite/hypotenuse. The adjacent side, by the Pythagorean theorem, is:

cosθ √(1 - x2)

Thus, the solution to the integral is:

(3/2)(arcsin(x) - x√(1 - x2)) C

Where C is the constant of integration.

Conclusion

By following the steps of substitution, simplification, and integration, we successfully solved the integral of 3x2 / √(1-x2). This problem exemplifies the power of trigonometric substitution in simplifying complex integrals. Remember, such substitutions can transform seemingly intractable problems into more manageable ones, making the world of calculus a bit more approachable.

Frequently Asked Questions

Q1: Can I use the same substitution technique for other integrals?

A1: Yes, trigonometric substitution is a versatile technique that can be applied to a variety of integrals, especially those involving square roots of quadratic expressions.

Q2: How do I determine which trigonometric substitution to use?

A2: The choice of substitution often depends on the form of the integral. For integrals involving √(a2 - x2), use x a sinθ; for √(a2 x2), use x a tanθ; and for √(x2 - a2), use x a secθ.

Q3: Is there a more efficient way to solve this integral than trigonometric substitution?

A3: While other methods might exist, trigonometric substitution often provides a structured approach to solve such integrals, making it a reliable and time-tested technique for mathematics students and professionals.