Integration of 1/(1 tan(3x)) Using a Trigonometric Substitution

When tackling integral problems involving trigonometric functions, a well-planned substitution can significantly simplify the process. This article will guide you through the method of solving ∫ 1/(1 tan(3x)) using a trigonometric substitution. This technique involves expressing the original trigonometric function in terms of a new variable to make the integration more manageable.

Introduction to the Problem

The integral to be solved is ∫ 1/(1 tan(3x)). The trigonometric identity and substitution are critical to resolving this problem effectively. By employing appropriate trigonometric identities, we can transform the integral into a form that can be solved using standard integration techniques.

Trigonometric Substitution and Simplification

Firstly, let’s use a substitution to simplify the integral. We set t tan(1.5x). This substitution utilizes the fact that tan(3x) tan(2 * 1.5x), which is a useful identity in simplifying the expression.

The derivative of the substitution, dt (3/2) * sec^2(1.5x) * dx, can be used to transform the original integral. We rewrite the tangent function in terms of the new variable t:

tan(3x) 2t / (1 - t^2)

Substituting these into the integral gives:

∫ 1 / (1 tan(3x)) dx ∫ 1 / (1 2t / (1 - t^2)) * (2/3) * sec^2(1.5x) dx

Partial Fraction Decomposition

The integral can now be decomposed using partial fractions. The expression 1 / (1 2t / (1 - t^2)) can be simplified to:

1 / (1 2t / (1 - t^2)) (1 - t^2) / (1 - 2t t^2) (1 - t^2) / ((1 - t)^2)

This simplifies the integral to:

∫ (1 - t^2) / ((1 - t)^2) * (2/3) * sec^2(1.5x) dx

Using the half-angle identity, tan(3x/2) t and x (2/3) * arctan(t), we can further simplify and integrate the function:

∫ 1 / (1 tan(3x)) dx (2/3) * ∫ (1 - t^2) / (2t - t^2) dt

This can now be broken down into partial fractions:

(1 - t^2) / (2t - t^2) 1/2 * (1 - t) / (t^2 - 1) 1/2 * (t - 1) / (t^2 - 2t - 1)

Final Integration and Substitution

Each of these terms can be integrated separately:

(1/3) * ∫ (1 - t) / (t^2 - 1) dt - (1/3) * ∫ (1 - t) / (t^2 - 2t - 1) dt

These integrals can be solved using standard techniques such as logarithmic and inverse tangent functions:

1/3 * (tan^(-1)(t) - (1/2) * ln|t^2 - 1| - (1/2) * ln|t^2 - 2t - 1|) C

Substituting back for t tan(3x/2), we get:

(1/3) * (tan^(-1)(tan(3x/2)) - (1/6) * ln|tan^2(3x/2) - 1| - (1/6) * ln|tan^2(3x/2) - 2tan(3x/2) - 1|) C

Finally, simplifying the expression using trigonometric identities:

(1/3) * (3x/2) - (1/6) * ln|cos(3x) sin(3x)| C

Conclusion

In summary, the key steps in solving the integral ∫ 1/(1 tan(3x)) dx involve using a trigonometric substitution to simplify the original form. The subsequent application of partial fractions allows for the integration of the simpler resulting terms. This method provides a powerful tool for dealing with integrals involving complex trigonometric expressions.

Mastering these techniques not only helps in solving specific integral problems but also enhances the overall understanding of trigonometric functions and their integrations. By practicing similar problems, one can develop a deeper insight into the areas of mathematics where such methods are applicable.