Understanding the Integration of Sin x Cos x: Methods and Applications

Integrating sin x cos x often requires a solid understanding of trigonometric identities and substitution methods. This guide explains three distinct approaches to finding the integral of sin x cos x, primarily focusing on the sin 2x formula and substitution techniques.

Integration of Sin x Cos x Using the Sin 2x Formula

The most straightforward approach to integrating sin x cos x is through the sin 2x formula. This method leverages the well-known trigonometric identity:

Trigonometric Identity

Given the identity sin 2x 2 sin x cos x, the integral can be simplified as follows:

[∫ sin x cos x dx ∫ (1/2) sin 2x dx]

Next, we integrate sin 2x with respect to x and apply the basic integration formula:

[∫sin 2x dx -1/2 cos 2x]

Putting it all together, we get:

[∫ sin x cos x dx -1/4 cos 2x C]

Integration of Sin x Cos x by Substituting Sin x

Another effective strategy involves using substitution. By setting u sin x, we can simplify the integral:

Substitution

The derivative of sin x is cos x dx du, which can be rewritten as cos x dx du. Substituting these into the original integral, we get:

[∫ sin x cos x dx ∫ u du]

The integral of u is straightforward:

[∫ u du u^2/2 C]

Substituting back for u sin x gives:

[∫ sin x cos x dx 1/2 sin^2 x C]

Integration of Sin x Cos x by Substituting Cos x

A third technique involves substituting cos x. Using the identity dcos x/dx -sin x, we can set v cos x and -sin x dx dv. This gives:

Substitution Method

Substituting these into the integral yields:

[∫ sin x cos x dx ∫ -v dv]

The integral of -v is:

[∫ -v dv -v^2/2 C]

Finally, substituting back for v cos x results in:

[∫ sin x cos x dx -1/2 cos^2 x C]

Conclusion

The integration of sin x cos x can be approached in multiple ways, providing a rich exploration of integral calculus. Each method highlights the importance of trigonometric identities and substitution techniques in simplifying and solving integrals. Whether using the sin 2x formula or substitution methods, these techniques offer a comprehensive understanding of integral calculus.