We are tasked with evaluating the following integral using the substitution rule:

$I = \int_{0}^{\frac{\pi}{3}} \frac{\sin(x)}{\cos^4(x)} \, dx$

Step 1: Choose the substitution

Let's start by making a substitution to simplify the integral. Notice that the presence of both $\sin(x)$ and $\cos(x)$ suggests that a substitution involving $\cos(x)$ might be helpful.

Let: $u = \cos(x)$ Then: $du = -\sin(x) \, dx$ Thus, $\sin(x) \, dx = -du$.

Step 2: Change the limits

Since $u = \cos(x)$, we need to change the limits of integration according to the substitution. When $x = 0$, $u = \cos(0) = 1$. When $x = \frac{\pi}{3}$, $u = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$.

So, the new limits are from $u = 1$ to $u = \frac{1}{2}$.

Step 3: Rewrite the integral in terms of $u$

The integral now becomes:

$I = \int_{1}^{\frac{1}{2}} \frac{-1}{u^4} \, du$ This simplifies to:

$I = -\int_{1}^{\frac{1}{2}} u^{-4} \, du$

Step 4: Evaluate the integral

Now, integrate $u^{-4}$. The integral of $u^n$ is $\frac{u^{n+1}}{n+1}$ for $n \neq -1$.

For $n = -4$, we have:

$\int u^{-4} \, du = \frac{u^{-3}}{-3} = -\frac{1}{3u^3}$

Thus, the integral becomes:

$I = -\left[ -\frac{1}{3u^3} \right]_{1}^{\frac{1}{2}}$

Step 5: Apply the limits

Now, apply the limits of integration:

$I = \frac{1}{3} \left[ \frac{1}{u^3} \right]_{1}^{\frac{1}{2}}$

$I = \frac{1}{3} \left( \frac{1}{\left(\frac{1}{2}\right)^3} - \frac{1}{1^3} \right)$

$I = \frac{1}{3} \left( \frac{1}{\frac{1}{8}} - 1 \right)$

$I = \frac{1}{3} \left( 8 - 1 \right)$

$I = \frac{1}{3} \times 7 = \frac{7}{3}$

Final Answer:

The value of the integral is:

$\boxed{\frac{7}{3}}$

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Would you like further clarification on any step, or have any additional questions?

Related Questions:

1. What would happen if we chose a different substitution, such as $u = \sin(x)$?
2. How do we handle integrals involving trigonometric identities in substitution problems?
3. What is the method for solving integrals of the form $\int \frac{\sin(x)}{\cos^n(x)} \, dx$?
4. How does the chain rule relate to integration by substitution?
5. Can you solve a similar integral where the powers of cosine or sine are different?

Tip:

Always carefully check how the limits of integration transform when performing substitution!