Problem 44 - Calculus: One and Several Variables

Question

Calculate. (If you run out of ideas, use the examples as models.) $$\int_{0}^{\pi / 3} \tan x \sec ^{3 / 2} x d x$$.

Step-by-Step Solution

Answer

A short version of the answer: First, apply the substitution \(u = \tan{x}\), then express the integral in terms of \(u\). The integral becomes: $$\int_{0}^{\sqrt{3}} (1 + u^2)^{3/4} du$$ Use the substitution rule again with \(v = 1 + u^2\), then use another substitution with \(w = \sqrt{v-1}\). The integral becomes: $$\int_{0}^{\sqrt{3}}w^2(w^2+1)^{3/4}dw$$ At this point, we need to use numerical methods or a calculator to get a final answer. This setup and understanding may be enough for the exercise requirements, depending on the context.

Step 1: Applying the substitution \(u = \tan{x}\)

To make the given integral easier to solve, let's define a new variable \(u\) like this: $$u = \tan{x}$$ $$ du = \frac{d(u)}{dx} = \sec^2{x} dx $$

Step 2: Expressing the integral in terms of \(u\)

Now, we can rewrite our integral in terms of \(u\). Using the relationship \(\sec^2{x} = 1 + \tan^2{x}\) and substituting the values we found in step 1, we get: $$\sec ^{3 / 2} x = (\sec^2{x})^{3/4} = (1+\tan^2{x})^{3/4} = (1+u^2)^{3/4}$$ Also, $$ dx = \frac{du}{\sec^2{x}} = \frac{du}{1 + u^2} $$ So now, we can rewrite the integral as $$\int_{0}^{\pi / 3} \tan x \sec ^{3 / 2} x d x = \int_{0}^{\sqrt{3}} u (1 + u^2)^{3/4} \frac{du}{1+u^2}$$ Since we changed variables, we must also change the limits of integration. When x=0, we have u=0 and when x=\(\frac{\pi}{3}\), u=\(\tan{\frac{\pi}{3}}=\sqrt{3}.\) Hence, the new integral becomes: $$\int_{0}^{\sqrt{3}} u (1 + u^2)^{3/4} \frac{du}{1+u^2}$$

Step 3: Simplify and evaluate the integral

Notice that the term \(\frac{u}{1 + u^2}\) in the integral cancels out. Now we can evaluate the integral: $$\int_{0}^{\sqrt{3}} (1 + u^2)^{3/4} du$$ To solve this integral, we can use the substitution rule again. Let's define a new variable, \(v\), like this: $$ v = 1 + u^2 $$ $$ \frac{dv}{du} = 2u $$ $$ du = \frac{dv}{2u} $$ When u=0, we have v=1 and when u=\(\sqrt{3}\), v=1+3=4. Thus, the new integral becomes: $$\int_{1}^{4} v^{3/4} \frac{dv}{2u}$$ To find u in terms of v, we can solve the equation \(v = 1 + u^2\) for u: $$u = \sqrt{v - 1}$$ Now, our integral is: $$\int_{1}^{4} v^{3/4} \frac{dv}{2\sqrt{v-1}}$$ This integral simplifies to: $$\frac{1}{2}\int_{1}^{4} (v-1)^{-1/2}v^{3/4} dv$$ Now, let's make another substitution: \(w = \sqrt{v-1}\), \(\frac{dw}{dv} = \frac{1}{2\sqrt{v-1}}\). Thus, \(dv = 2\sqrt{v-1} dw\). When v=1, w=0, and when v=4, w=\(\sqrt{3}\). Now, our integral is: $$\frac{1}{2}\int_{0}^{\sqrt{3}} w {(w^2+1)}^{3/4}2w dw$$ Which simplifies to: $$\int_{0}^{\sqrt{3}}w^2(w^2+1)^{3/4}dw$$ At this point, the integral becomes challenging to evaluate using elementary methods, and so we have to resort to numerical methods or a calculator to get a final answer. However, following the steps outlined here will give a student a solid understanding and setup of the problem. Depending on the context, this may be enough for the exercise requirements.

Key Concepts

Integration by Substitution

Integration by substitution is a powerful tool in calculus that allows us to simplify complex integrals. It's much like solving puzzles by changing pieces, making it easier to find the solution. Start by identifying a function inside your integral that can be replaced with a new variable, typically denoted as \(u\). This step is crucial to making the integral look simpler. For example, if you're working with \(\int \tan x \sec^{3/2} x \, dx\), you might substitute \(u = \tan x\). This substitution simplifies the original problem because it transforms the integrand into a simpler form.

Here's why substitution works:

• It reduces the complexity by turning a complicated expression into something simple.
• The derivative of your chosen substitution function (like \(du = \sec^2 x \, dx \) in the example) will appear elsewhere in the integral, allowing for a straightforward transformation.

Remember to adjust the limits of integration when performing definite integrals. After finding \(u\), express the differential \(dx\) in terms of \(du\), and change your limits accordingly to complete the transformation.
In essence, substitution allows you to turn an intimidating integral into a more manageable problem, playing a key role in solving integrals efficiently.

Trigonometric Integrals

Trigonometric integrals involve expressions with sine, cosine, tangent, and other trigonometric functions. These integrals often require unique approaches due to the oscillating nature of trig functions. They might seem daunting initially, but breaking them down makes them manageable. When dealing with trigonometric integrals, sometimes a direct approach using identities can help. For instance, knowing that \(\sec^2 x = 1 + \tan^2 x\) can be very useful.

In this specific exercise, we're facing an integral of the form \(\tan x \sec^{3/2} x\). Here are some strategies you can use:

• Utilize identities to express the integrand in a simpler form, as done by rewriting \(\sec^{3/2} x\) based on \(\sec^2 x\).
• Substitution methods can help simplify the trigonometric functions by turning them into polynomial-like expressions.

By making these transformations, evaluating trigonometric integrals can often become a straightforward task. Consider opting for more substitutions if necessary until reaching an easily solvable form. Although trig integrals may initially appear complex, breaking them down piece-by-piece, and using substitutions is the best way forward.

Limits of Integration

In calculus, especially when dealing with definite integrals, considering the limits of integration is crucial. When you transform your original variable to a new substitution variable, you must also transform the limits of integration accordingly.

This concept ensures you are evaluating the integral over the correct interval. For instance, initially, you might have limits derived from the variable \(x\), such as \(x = 0\) to \(x = \frac{\pi}{3}\). However, after substituting \(u = \tan x\), the limits change based on the new variable \(u\), giving you \(u = 0\) to \(u = \sqrt{3}\).

Keep these points in mind:

• Always adjust your integral limits when switching variables.
• Calculate the new limits using the substitution relationship.

This practice ensures that the integral respects the original boundaries of the problem. Without updating the limits of integration, you could end up evaluating an integral outside its intended scope. Therefore, always remember to adjust and evaluate these limits accurately.