91Ó°ÊÓ

The substitution method is a fundamental technique in calculus that simplifies integrals by changing variables. Here’s the idea: when you recognize that part of the integral is a function whose derivative also appears in the integral, you can make a substitution to simplify the integral into a more manageable form.

In the given exercise, we identified the substitution by setting \( u = \cos x \). This choice transformed the integral into a simpler expression. When we do this substitution, we must also express \( dx \) in terms of \( du \), which is achieved by differentiating \( u = \cos x \), giving us \( du = -\sin x \, dx \). Hence, \( -du = \sin x \, dx \), which fits perfectly into our integral form.

Here’s how it works in steps:

• Identify the part of the integral that can be substituted, ideally where its derivative is also present elsewhere in the integral.
• Make the substitution and also find an expression for \( dx \) in terms of \( du \).
• Rewrite the integral in terms of \( u \), eliminating \( x \) from the integral.

This method greatly facilitates solving challenging integrals by leveraging simpler integration techniques like the power rule.

The power rule for integration is an extremely useful technique for solving integrals of the form \( \int x^n \, dx \), where \( n eq -1 \). It provides a straightforward way to find antiderivatives by increasing the power of \( x \) by one and then dividing by this new power.

In our solution, the integration became \( \int u^{1/5} \, du \) after substitution. According to the power rule:

• Increase the exponent by one: \( n + 1 = \frac{1}{5} + 1 = \frac{6}{5} \).
• The new integral becomes \( \frac{u^{6/5}}{6/5} + C \), simplifying to \( \frac{5}{6} u^{6/5} + C \).

This approach is particularly potent because it transforms what might seem complex polynomials in integrals into simple problems through algebraic manipulation. Once the computation is done, remember to substitute back into terms of the original variable, which in this case is \( x \).
Understanding and applying the power rule is crucial for tackling a wide variety of integrals efficiently.

Trigonometric integrals involve functions like \( \sin x \), \( \cos x \), and other trigonometric identities. Solving these integrals often requires ingenuity, involving substitutions or recognizing trigonometric identities.

In our exercise, \( \cos ^n x \sin x \, dx \) is a form of trigonometric integral. Strategic substitution was key here, which leveraged the derivative relationship between sine and cosine.

• Recognize trigonometric identities to simplify expressions.
• Use substitutions like \( u = \cos x \) to turn the integral into a basic polynomial form that is easier to handle.
• Apply techniques like the power rule post-substitution to evaluate the integral.

Trigonometric integrals highlight the beauty and utility of interconnecting calculus with trigonometry, thus further broadening solving techniques and understanding of mathematics.