If f(x) = e^(cos x) sin x for |x| ≤ 2 and 2 otherwise, then ∫_(-2)³ f(x) dx = (a) 0 (b) 1 (c) 2 (d)

Q: Why do we split the integral at $x=2$? Can't we just integrate the whole thing at once?

Great question! In calculus, we can only integrate a function if it is continuous or piecewise continuous over the interval. Since the definition of $f(x)$ changes abruptly at $x=2$, we cannot apply the Fundamental Theorem of Calculus across that boundary in one go. Splitting the integral allows us to handle each piece with the appropriate definition, turning one complex problem into two manageable ones.

Q: I tried to integrate $e^{\cos x} \sin x$ using substitution, but it got messy. Is there a faster way?

You are absolutely right to feel that way! Attempting to find the antiderivative of $e^{\cos x} \sin x$ is a trap. While you *could* use substitution ($u = \cos x$), it is unnecessary here. Always pause and look at the limits of integration first. Symmetric limits like $[-2, 2]$ are a huge hint to check for odd/even properties before doing any heavy lifting.

Q: Wait, is $e^{\cos x} \sin x$ really an odd function? I thought the $e^{\cos x}$ part would mess it up.

That is a very common doubt! Remember, a function $g(x)$ is odd if $g(-x) = -g(x)$. Here, $g(-x) = e^{\cos(-x)} \sin(-x)$. Since $\cos(-x) = \cos x$, the exponential part $e^{\cos x}$ remains unchanged. Only the $\sin x$ part changes sign. Thus, the entire product becomes $-e^{\cos x} \sin x$, which is $-g(x)$. The exponential part doesn't 'mess it up'; it just scales the odd function, preserving its odd parity!

Question

If $f(x) = \begin{cases} e^{\cos x} \sin x, & 	ext{for } |x| \le 2 \ 2, & 	ext{otherwise} \end{cases}$ then $\int_{-2}^{3} f(x) dx = $

Options:

(A) 0

(B) 1

(C) 2

(D) 3

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with evaluating the definite integral $\int_{-2}^{3} f(x) \, dx$, where the function $f(x)$ is defined piecewise as:

$$f(x) = 
\begin{cases} 
e^{\cos x} \sin x & 	ext{for } |x| \le 2 \
2 & 	ext{otherwise}
\end{cases}$$

At first glance, this might look like a daunting task, but let's break it down into a logical journey.

### The Strategy of Splitting

The first thing to notice is the "seam" in our function. The definition changes exactly at $x = 2$.

In the world of integration, when a function behaves differently in different regions, we must respect those boundaries. We use the additive property of definite integrals:

$$\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx$$

By splitting our integral at $x = 2$, we transform one complex problem into two distinct, manageable ones:

$$\int_{-2}^{2} e^{\cos x} \sin x \, dx + \int_{2}^{3} 2 \, dx$$

### The Power of Symmetry

Now, look at the first integral: $I_1 = \int_{-2}^{2} e^{\cos x} \sin x \, dx$. Whenever you see symmetric limits like $[-a, a]$, your internal alarm bells should ring!

This is a massive hint to check for the parity of the function. Let $g(x) = e^{\cos x} \sin x$. To check if it is odd or even, we evaluate $g(-x)$.

Using the fundamental trigonometric identities, we know $\cos(-x) = \cos x$ and $\sin(-x) = -\sin x$. Substituting these, we get:

$$g(-x) = e^{\cos x} (-\sin x) = -g(x)$$

This confirms that $g(x)$ is an **odd function**. The beauty of an odd function over a symmetric interval is that the area below the $x$-axis perfectly cancels the area above it.

Thus, $\int_{-2}^{2} e^{\cos x} \sin x \, dx = 0$. The entire first part of our problem vanishes, leaving us with a much simpler task.

### The Geometric Conclusion

We are left with the second integral: $I_2 = \int_{2}^{3} 2 \, dx$. This is the integral of a constant, which is geometrically equivalent to finding the area of a rectangle.

The height of this rectangle is $2$, and the width is the interval length, $3 - 2 = 1$. Calculating the integral, we get:

$$[2x]_{2}^{3} = 2(3) - 2(2) = 6 - 4 = 2$$

Adding our two results together, $0 + 2 = 2$, we arrive at our final answer.

The final result is **2**. This problem is a masterclass in observation. By identifying the piecewise boundary and the symmetry of the integrand, we bypassed complex integration techniques entirely.

Animated Solution for Mathematics - Definite Integration: If $f (x) = {e^{c o s x} sin x, 2, for ∣ x ∣ \leq 2 otherwise$ then $\int_{- 2}^{3} f (x) d x =$

Select Answer:

Visualized Solution

Analyzing the Piecewise Function

Splitting the Integral

Setting Up the Split Integrals

Substituting the Function Definitions

Focusing on the First Integral

The Odd/Even Function Property

Checking Parity: Substituting $- x$

Simplifying the Trigonometric Terms

Concluding the Parity

Focusing on the Second Integral

Integrating the Constant

Applying the Limits

Calculating the Second Integral

Final Answer

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

The Strategy of Splitting

The Power of Symmetry

The Geometric Conclusion

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