91Ó°ÊÓ

When encountering integrals in calculus, particularly those involving composite functions or products of functions, u-substitution is a valuable tool for simplifying the expression into a more manageable form. It's similar to applying a change of variables in algebra but within the context of integration.

In the given exercise, we employ u-substitution by setting \( u = x^2 - \text{\pi} \), simplifying the integral by removing the composite trigonometric functions and turning it into an antiderivative involving \( u \). As we make this substitution, it's also crucial to convert \( dx \) into terms of \( du \), which is achieved by differentiating the inside function—yielding \( du = 2x dx \). This substitution, when applied correctly, allows us to rewrite the original integral in terms of \( u \), making it far easier to evaluate.

Integrals containing trigonometric functions, known as trigonometric integrals, can often be intricate to solve. Methods such as substitution, trigonometric identities, and sometimes integration by parts are used to tackle them.

The provided exercise involves an integral with both sine and cosine functions, compounded by the presence of an exponent. To address this, we apply a trigonometric identity, particularly the double-angle formula, which simplifies expressions involving products of sine and cosine.

Applying Trigonometric Identities

By using the identity \( 2\sin(u)\cos(u) = \sin(2u) \), we effectively reduce the complexity of the integral. In our case, the resultant integral involved a cube of sine function multiplied by a sine function with double the angle, further illustrating the usefulness of trigonometric identities in simplifying such expressions.

The power-reduction identity is pivotal in integral calculus, especially when dealing with higher powers of sine or cosine functions. The power-reduction formulas stem from the Pythagorean identity and help to express trigonometric functions of even powers in terms of the first power.

In the exercise, after applying u-substitution and trigonometric identities, we encounter an integral involving \( \sin^{2}(v) \). The power-reduction identity relevant here is \( \sin^{2} \theta = \frac{1 - \cos(2\theta)}{2} \), which allows us to rewrite the square of the sine function as a function of a double angle. This identity simplifies the original integral \( \int \sin^{4}(u)\cos(u) du \) into terms that are more straightforward to integrate.