Question

The value of \(\displaystyle\int^{10}_{-10}\frac{x^{10}\sin x}{\sqrt{1+x^{10}}}dx\) is equal to

• A. \(\frac{1}{100}\)
• B. \(\frac{-1}{100}\)
• C. \(\frac{1}{50}\)
• D. \(\frac{-1}{50}\)
• E. 0

Answer 1

Answer: (E)

We are tasked with evaluating the integral: \[ I = \int_{-10}^{10} \frac{x^{10} \sin x}{\sqrt{1 + x^{10}}} \, dx \] We observe that the integrand involves an odd function. Let’s examine the integrand for symmetry: \[ f(x) = \frac{x^{10} \sin x}{\sqrt{1 + x^{10}}} \] To check if the function is odd, we evaluate \(f(-x)\): \[ f(-x) = \frac{(-x)^{10} \sin(-x)}{\sqrt{1 + (-x)^{10}}} = \frac{x^{10} (-\sin x)}{\sqrt{1 + x^{10}}} = -\frac{x^{10} \sin x}{\sqrt{1 + x^{10}}} = -f(x) \] Since \( f(x) \) is an odd function, and we are integrating over a symmetric interval \([-10, 10]\), the integral of an odd function over a symmetric interval is zero. Therefore: \[ I = \int_{-10}^{10} f(x) \, dx = 0 \]

The correct option is (E) : \(0\)

Answer 2

We are given the integral:

\[ \int_{-10}^{10} \frac{x^{10} \sin x}{\sqrt{1 + x^{10}}} \, dx \]

Let us consider the **nature of the integrand**.

Define: \[ f(x) = \frac{x^{10} \sin x}{\sqrt{1 + x^{10}}} \]
Now, observe the function under \( f(-x) \):
\[ f(-x) = \frac{(-x)^{10} \sin(-x)}{\sqrt{1 + (-x)^{10}}} = \frac{x^{10} (-\sin x)}{\sqrt{1 + x^{10}}} = -\frac{x^{10} \sin x}{\sqrt{1 + x^{10}}} = -f(x) \]
Since \( f(-x) = -f(x) \), the function is odd.

And we are integrating an odd function over a symmetric interval \([-10, 10]\), so:
\[ \int_{-10}^{10} f(x) \, dx = 0 \]

Correct answer: 0