Question

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning. $$ \int \sin ^{456} x \cos x d x \text { or } \int \sin ^{2} x \cos ^{2} x d x $$

Short answer

\( \int \sin^2 x \cos^2 x \, dx \) is more difficult.

Step-by-step solution

Analyze Integral \( \int \sin^{456} x \cos x \, dx \)

To evaluate the integral \( \int \sin^{456} x \cos x \, dx \), one might consider using substitution. Let \( u = \sin x \), hence \( du = \cos x \, dx \). This transforms the integral into \( \int u^{456} \, du \), which is a straightforward power rule integral.

Analyze Integral \( \int \sin^2 x \cos^2 x \, dx \)

Evaluating \( \int \sin^2 x \cos^2 x \, dx \) requires using trigonometric identities to simplify the expression. One can use the double angle identity: \( \sin^2 x \cos^2 x = \frac{1}{4} \sin^2(2x) \). This transforms the integral into \( \frac{1}{4} \int \sin^2(2x) \, dx \), which involves using a reduction formula or half-angle identities, making it more complex.

Conclusion on Difficulty

The integral \( \int \sin^{456} x \cos x \, dx \) involves a simple substitution and a power rule, making it relatively straightforward. The integral \( \int \sin^2 x \cos^2 x \, dx \) requires trigonometric identities and additional steps, which increases its complexity. Therefore, \( \int \sin^2 x \cos^2 x \, dx \) is more difficult to evaluate.

Key concepts

Trigonometric Integrals

Trigonometric integrals are those that involve trigonometric functions like sine or cosine in the integrand. These types of integrals can range from relatively simple to highly complex, depending on the powers or combinations of the trigonometric functions involved. When evaluating trigonometric integrals, it's often useful to consider the structure of the integrand. For instance, if the integral is in the form of one trigonometric function to a high power, such as \( \int \sin^{456} x \, dx \), it might be solvable by straightforward methods. In contrast, integrals involving products of trigonometric functions with different powers, such as \( \int \sin^2 x \cos^2 x \, dx \), may require different techniques like trigonometric identities or substitutions to simplify them. In problems like these, recognizing patterns and deciding which techniques to apply can significantly simplify the integration process, making these integrals both a practice in algebraic manipulation and strategic decision-making.

Substitution Method

The substitution method is a powerful technique used to simplify integrals by transforming them into a format that is easier to solve. This approach involves substituting a part of the integrand with a new variable, which simplifies the integration process. For example:

• Consider the integral \( \int \sin^{456} x \cos x \, dx \). By letting \( u = \sin x \), we find \( du = \cos x \, dx \).


• This substitution simplifies the integral to \( \int u^{456} \, du \), which is a simple power integral that can be solved directly.

The fundamental goal of substitution is to convert the integral into one that is easier to solve, often reducing it to a basic form where standard integration techniques can be applied. This technique is particularly useful when dealing with trigonometric integrals where one function is the derivative of another.

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying integrals involving trigonometric functions. These identities allow us to rewrite expressions in a form that is easier to integrate. For instance:

• The double angle identity \( \sin(2x) = 2 \sin x \cos x \) can be manipulated to help integrate products of sine and cosine, like \( \sin^2 x \cos^2 x \).


• Given \( \sin^2 x \cos^2 x = \frac{1}{4} \sin^2(2x) \), the integral simplifies to \( \frac{1}{4} \int \sin^2(2x) \, dx \).

Using identities can sometimes introduce new integrals that may still require additional techniques to solve, such as half-angle formulas or reduction formulas. This makes solving some trigonometric integrals more involved, especially when the identity introduces expressions that are more complex than anticipated. It’s about finding the right identity that can simplify the integral to its core.