Solving The Integral: E^x(1 - Cot X + Cot^2 X) Dx

Hey math enthusiasts! Today, we're diving into a calculus problem that looks a bit intimidating at first glance, but with a little bit of clever manipulation, we can totally crack it. Our mission, should we choose to accept it, is to solve the integral of $\int e ^x\left(1-\cot x+\cot ^2 x\right) d x$. Don't worry, we'll break it down step by step, so even if you're not a math whiz, you'll be able to follow along. We'll explore different approaches, simplify the expression, and ultimately arrive at the correct answer. Let's get started, shall we?

Understanding the Problem: The Integral's Components

Alright, guys, let's take a closer look at what we're dealing with. The integral is $\int e ^x\left(1-\cot x+\cot ^2 x\right) d x$. The main player here is the function inside the integral, which is $e^x(1 - ext{cot }x + ext{cot}^2 x)$. The presence of $e^x$ suggests that integration by parts might be a useful technique down the line, but first, let's see if we can simplify the expression inside the parentheses. We have a constant term (1), a cotangent term ($- ext{cot }x$), and a cotangent squared term ($ ext{cot}^2 x$). Remember, the goal is to find the antiderivative of this function. To do this, we need to find a function whose derivative is equal to the given function. This might involve applying integration rules, trigonometric identities, and algebraic manipulations.

So, what strategies can we use? We can try to recognize some patterns. For instance, the derivative of $ ext{cot }x$ is $- ext{csc}^2 x$. Also, we know the trigonometric identity $1 + ext{cot}^2 x = ext{csc}^2 x$. This is super handy! We can rewrite the expression inside the parentheses using this identity to see if we can simplify things. Sometimes, just rearranging terms can help you spot a clever trick to solve the problem. Remember, in calculus, creativity is key, and often, there's more than one way to get to the answer. Let's see how far we can get with our mathematical tools!

Strategic Simplification: Using Trigonometric Identities

Now, let's flex those math muscles and simplify our integral. Remember our expression: $\int e ^x\left(1-\cot x+\cot ^2 x\right) d x$. Our goal here is to make it easier to integrate. The key lies in recognizing that $1 + ext{cot}^2 x = ext{csc}^2 x$. Let's rewrite the expression inside the parentheses: $1 - ext{cot }x + ext{cot}^2 x$. We can rearrange this as follows: $(1 + ext{cot}^2 x) - ext{cot }x$. And now, using the trigonometric identity, we get $ extcsc}^2 x - ext{cot }x$. So, our integral now looks like this $\int e^x ( ext{csc^2 x - ext{cot }x) dx$. This is much better, as the derivative of $ ext{cot }x$ is $- ext{csc}^2 x$. This is a big clue!

This rearrangement is the key to solving the integral. The derivative of $ ext{cot }x$ is $- ext{csc}^2 x$. Therefore, if we apply integration by parts, we might be able to find a way to eliminate one of the terms and simplify the overall expression. Let's start by rewriting our integral. Our integral is of the form $\int e^x ( ext{csc}^2 x - ext{cot }x) dx$. We can rewrite this as $\int e^x ext{csc}^2 x dx - \int e^x ext{cot }x dx$. Now, here is where we need to remember the integration by parts formula: $\int u dv = uv - \int v du$. By strategically choosing $u$ and $dv$, we can simplify the integral. Sometimes, it is necessary to consider alternative methods of integration. Maybe another substitution is required, or a different trigonometric identity might prove helpful in simplifying the overall expression. Keep in mind that the best method may not always be immediately apparent.

Applying Integration by Parts: The Road to the Solution

Alright, folks, it's time to put our integration by parts skills to the test. Recall the integral: $\int e^x ( ext{csc}^2 x - ext{cot }x) dx$. We'll focus on the two parts: $\int e^x ext{csc}^2 x dx - \int e^x ext{cot }x dx$. Let's see if we can use integration by parts strategically here. We can rewrite the integral as: $\int e^x ext{csc}^2 x dx - \int e^x ext{cot }x dx$. The derivative of $ ext{cot }x$ is $- ext{csc}^2 x$. This gives us a hint! Let's apply integration by parts to the term $ \int e^x \cot x dx$.

Let's apply integration by parts to the term $\int e^x \cot x dx$. We'll let $u = ext{cot }x$ and $dv = e^x dx$. Then, $du = - ext{csc}^2 x dx$ and $v = e^x$. Using the integration by parts formula: $\int u dv = uv - \int v du$, we get:

$\int e^x \cot x dx = e^x \cot x - \int e^x (- ext{csc}^2 x) dx$ $\int e^x \cot x dx = e^x \cot x + \int e^x \text{csc}^2 x dx$

Now, let's substitute this back into our original integral: $\int e^x ( ext{csc}^2 x - ext{cot }x) dx = \int e^x ext{csc}^2 x dx - (e^x \cot x + \int e^x \text{csc}^2 x dx)$. This simplifies to: $\int e^x ext{csc}^2 x dx - e^x \cot x - \int e^x \text{csc}^2 x dx$. Notice something amazing? The $\int e^x \text{csc}^2 x dx$ terms cancel out! This leaves us with: $-e^x \cot x$. And there you have it, guys!

The Final Answer and Verification

So, after all that work, we've found that $\int e ^x\left(1-\cot x+\cot ^2 x\right) d x = -e^x \cot x + C$, where C is the constant of integration. Therefore, the correct answer from the options you provided is (b) $- e ^x \cot x$. To be absolutely sure, we can take the derivative of our answer, $-e^x \cot x$, and see if it matches the original integrand, $e^x(1 - ext{cot }x + ext{cot}^2 x)$.

Let's use the product rule. The derivative of $-e^x \cot x$ is:

$-(e^x \cdot \cot x + e^x \cdot (-\text{csc}^2 x))$ $=-e^x \cot x + e^x \text{csc}^2 x$ $= e^x (\text{csc}^2 x - \cot x)$

Using the identity $1 + \text{cot}^2 x = \text{csc}^2 x$, we can rewrite this as:

$= e^x (1 + \text{cot}^2 x - \cot x)$ $= e^x (1 - \text{cot }x + \text{cot}^2 x)$

And there you have it! The derivative of our answer matches the original integrand, confirming that our solution is correct. Isn't math satisfying?

Key Takeaways and Conclusion

So, what did we learn from this problem, guys? Well, first off, we saw how crucial it is to recognize trigonometric identities. Knowing that $1 + \text{cot}^2 x = \text{csc}^2 x$ was the game-changer here. Second, integration by parts is a powerful tool, but it's essential to apply it strategically. Choosing the right $u$ and $dv$ can make all the difference. And third, don't be afraid to rearrange and simplify! Sometimes, a little bit of algebraic manipulation can reveal a much easier path to the solution. In summary, we have successfully solved the integral $\int e ^x\left(1-\cot x+\cot ^2 x\right) d x$ and confirmed that the correct answer is (b) $- e ^x \cot x$. Congratulations! You have successfully navigated through a calculus problem. Keep practicing, keep exploring, and keep the math love alive! Until next time, keep calculating!