Question

The values of the following integrals are known and can be found in integral tables or by computer. Your goal in evaluating them is to learn about contour integration by applying the methods discussed in the examples above. Then check your answers by computer. $${\int }_0^{\mathrm{\infty }}\frac{x\sin xdx}{9x^2+4}$$

Short answer

$\frac{4\pi }{27}\exp (-3/2)$

Step-by-step solution

Identify the integral

The given integral is ${\int }_0^{\mathrm{\infty }}\frac{x\sin xdx}{9x^2+4}.$

Consider invoking integral tables

In integral tables, the integral can be found as a standard form. This specific form is ${\int }_0^{\mathrm{\infty }}\frac{x\sin (ax)dx}{x^2+b^2}=\frac{\pi }{2}\frac{a}{b^3}{e}^{-ab}.$

Adjust the formula to match the given integral

To match our given integral ${\int }_0^{\mathrm{\infty }}\frac{x\sin xdx}{9x^2+4},$ set $a=1$ and $b^2=9/4.$

Evaluate the expression

Using $b=\sqrt{9/4}=3/2,$ the integral becomes ${\int }_0^{\mathrm{\infty }}\frac{x\sin xdx}{9x^2+4}=\frac{\pi }{2}\frac{1}{(3/2{)}^3}{e}^{-(1)(3/2)}$ which simplifies to $\frac{\pi }{2}\frac{1}{27/8}{e}^{-3/2}.$

Simplify the result

Thus, the integral can be simplified to $\frac{\pi }{2}\frac{8}{27}{e}^{-3/2},$ or more compactly, $\frac{4\pi }{27}{e}^{-3/2}.$

Key concepts

Integral Tables

Integral tables are collections of integrals that have been previously solved and tabulated for easy reference. These tables are extremely helpful when you encounter complex integrals, such as ${\int }_0^{\mathrm{\infty }}\frac{x\sin xdx}{9x^2+4}$. Rather than solving these integrals from scratch, you can look them up in a table, match them to a known form, and use the result directly.
To use integral tables effectively:

• Identify the form of your given integral.
• Check if this form matches any standard integrals listed in the table.
• Adjust the parameters in the standard form to fit your specific integral.

In our exercise, we matched our integral to the form ${\int }_0^{\mathrm{\infty }}\frac{x\sin (ax)dx}{x^2+b^2}=\frac{\pi }{2}\frac{a}{b^3}{e}^{-ab}$. This allowed us to quickly evaluate the integral without going through complex calculations.

Integral Evaluation

Evaluating integrals is a fundamental skill in calculus. It involves finding the function that describes the area under the curve represented by the integral. In our example, we evaluated ${\int }_0^{\mathrm{\infty }}\frac{x\sin xdx}{9x^2+4}$ using both integral tables and manual calculations.
Here's the step-by-step process:

• First, we identified the integral's form.
• Next, we matched it to a standard form found in an integral table.
• We adjusted the parameters 'a' and 'b' to fit our specific integral.
• We then used the standard result provided by the table to evaluate the integral.

Finally, we simplified the result to obtain $\frac{4\pi }{27}{e}^{-3/2}$. This method not only simplifies the calculation but also reinforces understanding of how standard forms can be adapted to find solutions.

Sinusoidal Integration

Integrating functions involving sine or cosine, like in our integral ${\int }_0^{\mathrm{\infty }}\frac{x\sin xdx}{9x^2+4}$, falls under sinusoidal integration. These integrals often appear in physics and engineering problems because sinusoidal functions describe wave patterns.
Key points to remember:

• Sinusoidal integrals often have forms that are tabulated for easy reference.
• The trigonometric identity used to simplify the integral depends on the function (sine or cosine).
• Applying known identities and transformations can significantly simplify the integral.

In our example, incorporating the standard sinusoidal integral form directly from the table helped us bypass more complex algebraic manipulations. The essential trick was recognizing that we had a sinusoidal function and leveraging that to match a known integral form.

Definite Integrals

Definite integrals, like the one we worked on ${\int }_0^{\mathrm{\infty }}\frac{x\sin xdx}{9x^2+4}$, compute the area under the curve over a specific interval. In this case, from 0 to $\mathrm{\infty }$. Definite integrals provide precise numerical values rather than an indefinite formula.
Understanding definite integrals involves:

• Recognizing the limits of integration (in our example, from 0 to $\mathrm{\infty }$).
• Applying integral rules and transformations appropriately within these limits.
• Ensuring proper simplification and evaluation at the bounds defined.

In our supplied solution, the process included handling the infinite upper limit by recognizing known integral behaviors and leveraging results from integral tables. This made evaluating the definite integral straightforward and accurate.