Integration using Euler's formula

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In integral calculus, complex numbers and Euler's formula may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of eix and eix, and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts, and is sufficiently powerful to integrate any rational expression involving trigonometric functions.

Euler's formula[edit]

Euler's formula states that

Substituting x for x gives the equation

These two equations can be solved for the sine and cosine:

Simple example[edit]

Consider the integral

The standard approach to this integral is to use a half-angle formula to simplify the integrand. We can use Euler's identity instead:

At this point, it would be possible to change back to real numbers using the formula e2ix + e−2ix = 2 cos 2x. Alternatively, we can integrate the complex exponentials and not change back to trigonometric functions until the end:

Second example[edit]

Consider the integral

This integral would be extremely tedious to solve using trigonometric identities, but using Euler's identity makes it relatively painless:

At this point we can either integrate directly, or we can first change the integrand to cos 6x − 2 cos 4x + cos 2x and continue from there. Either method gives

Using real parts[edit]

In addition to Euler's identity, it can be helpful to make judicious use of the real parts of complex expressions. For example, consider the integral

Since cos x is the real part of eix, we know that

The integral on the right is easy to evaluate:

Thus:

Fractions[edit]

In general, this technique may be used to evaluate any fractions involving trigonometric functions. For example, consider the integral

Using Euler's identity, this integral becomes

If we now make the substitution u = eix, the result is the integral of a rational function:

Any rational function is integrable (using, for example, partial fractions), and therefore any fraction involving trigonometric functions may be integrated as well.