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Chapter 7: Problem 46

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

Short Answer

Expert verified
In order to derive the reduction formula for the integral of x^n * sin(ax) with respect to x, we first use integration by parts and set u = x^n and dv = sin(ax) dx. Next, we find du and v by differentiating u and integrating dv, respectively. After substituting the expressions of u, v, du, and dv into the integration by parts formula, we simplify the integral. The final reduction formula is given by: $$\int x^{n} \sin a x d x=-\frac{x^{n} \cos a x}{a}+\frac{n}{a} \int x^{n-1} \cos a x d x, \quad \text { for } a \neq 0$$.

Step by step solution

01

Find du and v

We need to find du, which is the derivative of u with respect to x, and v, which is the integral of dv. $$u = x^n \implies du = nx^{n-1} \, dx$$ $$dv = \sin a x \, dx \implies v = \int \sin a x \, dx = -\frac{1}{a} \cos a x$$
02

Apply Integration by Parts

Now that we have u, v, du, and dv, we will apply integration by parts, using: $$\int u \, dv = uv - \int v \, du$$ Substituting the expressions for u, v, du, and dv yields: $$\int x^n \sin a x \, dx = -\frac{x^n \cos a x}{a} - \int -\frac{1}{a} nx^{n-1} \cos a x \, dx$$
03

Simplify the Integral

Now, let's simplify the integral: $$\int x^n \sin a x \, dx = -\frac{x^n \cos a x}{a} +\frac{n}{a} \int x^{n-1}\cos a x \, dx$$ We have successfully derived the reduction formula using integration by parts. The result is: $$\int x^{n} \sin a x d x=-\frac{x^{n} \cos a x}{a}+\frac{n}{a} \int x^{n-1} \cos a x d x, \quad \text { for } a \neq 0$$

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