Question

Use integration by parts to verify the reduction formula. $$ \int \sin ^{n} x d x=-\frac{\sin ^{n-1} x \cos x}{n}+\frac{n-1}{n} \int \sin ^{n-2} x d x $$

Short answer

The reduction formula given is valid. The process of confirming this involves applying the integration by parts formula, subsequently simplifying and rearranging terms. Eventually, this method verifies the given reduction formula.

Step-by-step solution

Identification

To use integration by parts, we first identify \( u \) and \( dv \), such that \( u = \sin^{n-1}x \) and \( dv = \sin x dx \). Then we need to find \( du \) and \( v \), \( du = (n-1)\sin^{n-2}x \cos x dx \) and \( v = -\cos x \).

Integration by Parts

We substitute \( u \), \( dv \), \( v \), \( du \) into the integration by parts formula: \[\int u dv = uv - \int v du\] This gives us: \[\int \sin^{n}x dx = -\sin^{n-1}x \cos x - \int -(n-1)\sin^{n-2}x \cos^2 x dx \] Remember that \( \cos^2 x = 1 -\sin^2 x \) thus the equation simplifies to \[\int \sin^{n}x dx = -\sin^{n-1}x \cos x +(n-1)\int \sin^{n-2}x dx + (n-1)\int \sin^n x dx\]

Simplifying

we'll have a \( \int \sin^n x dx \) on both sides of the equation. We should write them on the same side: \[n \int \sin^n x dx = -\sin^{n-1} x \cos x + (n-1)\int \sin^{n-2} x dx\]Then divide the entire equation by \( n \):\[\int \sin^n x dx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n}\int \sin^{n-2} x dx\]which verifies our initial formula.

Key concepts

Reduction Formula

Reduction formulas are powerful tools in calculus that allow for the simplification of certain types of integrals, particularly those involving powers of trigonometric functions. A reduction formula essentially breaks down a complex integral into a simpler form that's more manageable to solve. They are recursive in nature, expressing the integral of a function in terms of an integral with a lower power.

When faced with a challenging integral such as \[\int \sin^{n} x dx\], it can feel daunting due to the exponent on the sine function. However, by applying the reduction formula as shown in the exercise, we reduce the power of the sine function incrementally with each iteration. Not only does this make the problem easier to handle, but it also provides a systematic approach to integrating powers of trigonometric functions.

Using reduction formulas efficiently requires understanding the process of integration by parts and recognizing patterns within the formulas. It's crucial to identify the parts of the integrand that can be set as 'u' and 'dv' for the integration by parts to work. The goal is to simplify the integral step by step, repeating the process if necessary, until the integral is in a form that can be solved directly or recognized as a standard integral.

Integrating Trigonometric Functions

Trigonometric functions are common in calculus, and learning to integrate them is a fundamental skill. The challenge in integrating trigonometric functions like sine and cosine comes when these functions are raised to a power. Straightforward antiderivatives are not always evident, and that's where techniques such as substitution and integration by parts become invaluable.

In the context of our reduction formula example, integrating \(\sin^{n} x\) was accomplished by recognizing that integration by parts would allow us to express the integral in terms of a lower power of sine. The key is to select 'u' and 'dv' properly, as was done in the exercise, to simplify the integral to a basic form, or to another integral that is easier to evaluate.

Remember that trigonometric identities, like \(\cos^2 x = 1 - \sin^2 x\), are often used to transform the integral into a more solvable state. Being familiar with these identities and knowing when to apply them can make a significant difference in solving complex trigonometric integrals.

Calculus Problem Solving

Strategies for Approaching Calculus Problems

Recognizing Patterns: A critical skill in calculus problem solving is pattern recognition. By identifying similarities to previously solved problems, one can leverage known strategies, such as the application of reduction formulas or trigonometric identities like we saw in this example.

Understanding Concepts: Rather than memorizing formulas and techniques, truly understanding why and how they work arms you with the flexibility to tackle a wide range of problems. For instance, by understanding integration by parts conceptually, you're able to apply it to verify the reduction formula for \(\sin^{n} x\).

Practice and Repetition: Mastery in calculus comes with practice. Working through various problems enhances your ability to apply different integration techniques effectively and recognize when certain methods are best suited for the task.

Seeking Help and Using Resources: Don't hesitate to use educational resources, whether it's a textbook, online platforms, or assistance from a tutor. Modern educational platforms offer step-by-step solutions that provide valuable insights into calculus problem-solving processes.

The iterative approach of verifying the reduction formula using integration by parts in the given exercise is illustrative of how calculus problems often require multiple passes and techniques. Learning to creatively combine these tools can turn a seemingly intractable problem into a manageable one.