Question

Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral. $$ \int x^{n} \cos x d x=x^{n} \sin x-n \int x^{n-1} \sin x d x $$

Short answer

Use integration by parts: \(\int x^n \cos x \, dx = x^n \sin x - n \int x^{n-1} \sin x \, dx.\)

Step-by-step solution

Identify Functions for Integration by Parts

Recall the integration by parts formula: \(\int u\, dv = uv - \int v\, du\). We need to choose \(u\) and \(dv\) such that the integral becomes simpler. Here, choose \(u = x^n\) so that \(du = n x^{n-1} dx\), and \(dv = \cos x\, dx\) so that \(v = \sin x\).

Apply Integration by Parts Formula

Using the formula \(\int u\, dv = uv - \int v\, du\), substitute \(u = x^n\), \(du = n x^{n-1}\, dx\), \(dv = \cos x\, dx\), and \(v = \sin x\) into the formula: \[ \int x^n \cos x \, dx = x^n \sin x - \int (\sin x)(n x^{n-1}) \, dx. \]

Simplify the Expression

Simplify the integral on the right side: \[ \int n x^{n-1} \sin x \, dx = n \int x^{n-1} \sin x \, dx. \] Thus, the simplified expression is: \[ \int x^n \cos x \, dx = x^n \sin x - n \int x^{n-1} \sin x \, dx. \]

Final Formula

The reduction formula is derived as: \[ \int x^n \cos x \, dx = x^n \sin x - n \int x^{n-1} \sin x \, dx. \] This matches the expression given in the problem statement.

Key concepts

Reduction Formulas

Reduction formulas are a clever strategy used in integration to simplify a problem step by step. Imagine having a big puzzle to solve; instead of trying to solve it all at once, you break it into smaller, more manageable pieces. That's what reduction formulas do in integration.

When you encounter an integral that's difficult to evaluate directly, reduction formulas can help break it down. The given integral \( \int x^n \cos x \; dx \) is a perfect example. By using reduction formulas, you reduce the power of \( x \) in each step, making the integral easier to evaluate. The 'reduction' happens because the formula transforms the original problem into an easier one, as seen where the new integral, after one iteration, is \( \int x^{n-1} \sin x \; dx \).

Reduction formulas are especially useful for integrals involving powers of a variable and function combinations, such as trigonometric or exponential functions. Once you get the hang of this technique, handling integrals like this feels much more intuitive.

Calculus Integration Techniques

In calculus, integration techniques are your toolkit for solving different types of integrals. These techniques include things like substitution, integration by parts, and partial fraction decomposition.

For the given integral, integration by parts is especially handy. It's like solving a problem using clever teamwork. You break the integral into two parts—one you differentiate and one you integrate. This works best for products of functions, like \( x^n \) and \( \cos x \) in our example. Here, we choose \( u = x^n \) to differentiate, and \( dv = \cos x \; dx \) to integrate. This choice is crucial as it leads to a simpler integral.

Integration by parts is represented by the formula:

• \( \int u\, dv = uv - \int v\, du \)

By applying this method, the integral simplifies step by step. It's a powerful technique because it transforms the integral into one that's easier to solve, often involving more straightforward functions or lower powers.

Integration of Trigonometric Functions

Integration of trigonometric functions, like \( \cos x \), often leads to specific integral calculations that might seem challenging at first. However, with the right techniques, like integration by parts, it becomes much simpler. Trigonometric functions have specific patterns that, when identified, allow us to simplify the integration process.

In our integral, \( \int x^n \cos x \; dx \), handling the \( \cos x \) term requires knowing its antidifferentiation, which is \( \sin x \). This knowledge helps us pick derivatives and integrals neatly when using techniques like integration by parts.
When integrating trigonometric functions, remembering their basic derivatives and antiderivatives is crucial, as it guides your approach to find solutions efficiently. Moreover, many trigonometric integrals can be found in tables, which offer pre-evaluated solutions for common integral forms, saving time and effort.

• \( \int \cos x \; dx = \sin x + C \)
• \( \int \sin x \; dx = -\cos x + C \)

By combining this knowledge with integration techniques, particularly when dealing with polynomial and trigonometric product integrals, you're armed to tackle even complex-looking integrals with confidence.