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Chapter 7: Problem 45

Use integration by parts to derive the following reduction formulas. $$\int x^{n} \cos a x d x=\frac{x^{n} \sin a x}{a}-\frac{n}{a} \int x^{n-1} \sin a x d x, \quad \text { for } a \neq 0$$

Short Answer

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Question: Derive the reduction formula for the given integral using integration by parts: $$\int x^{n} \cos a x d x=\frac{x^{n} \sin a x}{a}-\frac{n}{a} \int x^{n-1}\sin a x d x, \quad \text { for } a \neq 0$$

Step by step solution

01

Choose functions u and dv

We will choose our functions u and dv as follows: $$u = x^n$$ $$dv = \cos ax dx$$ Now, we need to differentiate u and integrate dv to get du and v, respectively.
02

Differentiate u and integrate dv

First, let's differentiate u: $$du = \frac{d}{dx}(x^n) = nx^{n-1}dx$$ Next, let's integrate dv: $$v = \int \cos ax dx = \frac{1}{a}\sin ax + C$$
03

Apply the integration by parts formula

Using the integration by parts formula and our chosen functions: $$\int x^{n} \cos a x d x = uv - \int vdu$$ Now, substitute the values we found for u, v, and du: $$\int x^{n} \cos a x d x = \left(\frac{x^{n} \sin ax}{a}\right) - \int \left(\frac{1}{a}\sin ax\right)(nx^{n-1} dx)$$
04

Simplify the integral

We can simplify the integral as follows: $$\int x^{n} \cos a x d x = \frac{x^{n} \sin a x}{a} - \frac{n}{a}\int x^{n-1}\sin ax dx$$ This is the reduction formula that we set out to derive: $$\int x^{n} \cos a x d x = \frac{x^{n} \sin a x}{a}-\frac{n}{a} \int x^{n-1}\sin a x d x, \quad \text { for } a \neq 0$$

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