Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. $${\int }_0^4\frac{p}{\sqrt{9+p^2}}dp$$

Short answer

Question: Evaluate the definite integral: $${\int }_0^4\frac{p}{\sqrt{9+p^2}}dp$$ Answer: $$3\cosh (3\sinh (4))-3$$

Step-by-step solution

Choose a substitution

Let's choose a substitution that will simplify the expression under the square root. A useful substitution in this case is: $$u=3\sinh x$$ such that $$p=3\sinh x\Rightarrow p^2=9{\sinh }^{2}x$$ Now, we have to substitute $x$ for $p$ in the integral. Now we can differentiate $u$ with respect to $x$ to find: $$\frac{dp}{dx}=3\cosh x\Rightarrow dp=3\cosh xdx$$

Change the integral variable and limits

Now, we change the definite integral according to our substitution. Using the new variable, the integral becomes: $${\int }_{u(0)}^{u(4)}\frac{p}{\sqrt{9+p^2}}dp={\int }_{u(0)}^{u(4)}\frac{3\sinh x}{\sqrt{9+9{\sinh }^{2}x}}(3\cosh x)dx$$ Now we need to find the new limits of integration, which are $u(0)$ and $u(4)$: $$u(0)=3\sinh (0)=0$$ $$u(4)=3\sinh (4)$$ Now, we can rewrite the integral with the new limits: $${\int }_0^{3\sinh (4)}\frac{3\sinh x}{\sqrt{9+9{\sinh }^{2}x}}(3\cosh x)dx$$

Simplify the integral

Now we should try to simplify the integral. Notice that under the square root, we can factor out the 9 as follows: $$\sqrt{9+9{\sinh }^{2}x}=\sqrt{9(1+{\sinh }^{2}x)}=\sqrt{9{\cosh }^{2}x}=3\cosh x$$ Then, the integral simplifies to: $${\int }_0^{3\sinh (4)}\frac{3\sinh x}{3\cosh x}(3\cosh x)dx={\int }_0^{3\sinh (4)}3\sinh xdx$$

Evaluate the integral

Now, the integral is simple to solve. The antiderivative of $3\sinh x$ is $3\cosh x$. Thus, we evaluate the definite integral as follows: $${\int }_0^{3\sinh (4)}3\sinh xdx={[3\cosh x]}_0^{3\sinh (4)}=3\cosh (3\sinh (4))-3\cosh (0)=3\cosh (3\sinh (4))-3$$ So the definite integral evaluates to: $${\int }_0^4\frac{p}{\sqrt{9+p^2}}dp=3\cosh (3\sinh (4))-3$$

Key concepts

Change of Variables

In calculus, the method of a change of variables (also known as substitution) is a very useful technique for solving integrals, especially definite integrals. This technique involves replacing the original variable with a new one, simplifying the integral's computation. For example, converting the definite integral from a complex function of one variable to an easier one of another makes it easier to compute.

In the original exercise, the substitution of variables simplifies the integral significantly. By substituting $u=3\sinh x$, and consequently $p=3\sinh x$, we transformed the complex expression into a simpler hyperbolic function. Substitution here is just like translating a foreign language too complex to read at first glance into a more familiar script.

Key factors to consider during substitution:

• Choose a substitution that simplifies the expression, often targeting the most complicated part of the integrand.
• Differentiate your chosen substitution to express $dp$ or the original variable in terms of your new variable.
• Don’t forget to change the limits of integration accordingly based on your substitution.

Hyperbolic Substitution

Hyperbolic substitution is an unconventional, yet powerful, tool in integration, especially when dealing with expressions involving square roots. It involves using hyperbolic functions, which have similar properties to trigonometric functions, but are defined differently.

For instance, in the problem, we see the substitution $u=3\sinh x$. This substitution cleverly uses the identity $1+{\sinh }^{2}x={\cosh }^{2}x$ to simplify under the square root. By recognizing or applying these identities, the integral simplifies, making it possible to solve.

To get the most out of hyperbolic substitutions:

• Understand the basic hyperbolic identities and how they can help simplify square roots or other operations.
• Know the corresponding hyperbolic integral formulas, which are often necessary to complete the substitution process.
• Practice identifying situations where hyperbolic substitutions might simplify your problem.

Integration Techniques

Integration techniques are strategies used in calculus to evaluate integrals that are not straightforward. Understanding these techniques allows you to solve complex integrals by transforming them into simpler computations.

In our example, after substitution and manipulation of terms inside the integral, we used the property of hyperbolic identities to simplify the function before proceeding with the integration. These steps highlight the importance of:

• Recognizing opportunities to simplify the integrand.
• Selecting appropriate algebraic manipulations, such as factoring or applying identities.
• Executing a careful and systematic approach to eliminate complexities in the integral.

By practicing various methods like substitution, partial fraction decomposition, and recognizing standard integrals, you can expedite the process of finding solutions to integrals.

Limit Evaluation

In the context of definite integrals, evaluating limits is crucial. These limits define the range over which the function is integrated and are susceptible to change during variable substitution.

Once a substitution is made, as seen with $u=3\sinh x$, you must also transform the original limits of integration. By calculating $u(0)=0$ and $u(4)=3\sinh (4)$, it ensures accurate integration over the new variable’s appropriate range.

Consider these steps for proper limit evaluation:

• Always recompute the integration bounds after variable substitution to maintain the integrity of the problem.
• Substitute the original bounds into your substitution equation to find the new limits.
• Ensure accuracy in calculations to avoid errors in the final evaluation of the definite integral.