Question:

Step 1: Identify the integral. Let the given integral be $I$. $$ I = \int_{-1}^{1} x \sin(\pi x) \, dx $$ Step 2: Determine the symmetry of the integrand. Let $f(x) = x \sin(\pi x)$. To determine if $f(x)$ is an even or odd function, we evaluate $f(-x)$: $$ f(-x) = (-x) \sin(\pi (-x)) $$ Using the trigonometric identity $\sin(-\theta) = -\sin(\theta)$: $$ f(-x) = (-x) (-\sin(\pi x)) $$ $$ f(-x) = x \sin(\pi x) $$ Since $f(-x) = f(x)$, the integrand $f(x) = x \sin(\pi x)$ is an even function. Step 3: Apply the property of definite integrals for even functions over a symmetric interval. For a definite integral over a symmetric interval $[-a, a]$, if $f(x)$ is an even function, the integral can be expressed as: $$ \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx $$ In this case, $a=1$, and $f(x) = x \sin(\pi x)$ is an even function. Therefore, $$ I = 2 \int_{0}^{1} x \sin(\pi x) \, dx $$ Step 4: Evaluate the integral using integration by parts. We use the integration by parts formula: $\int u \, dv = uv - \int v \, du$. Let $u = x$ and $dv = \sin(\pi x) \, dx$. Then, $du = dx$ and $v = \int \sin(\pi x) \, dx = -\frac{\cos(\pi x)}{\pi}$. Applying the formula to the definite integral: $$ \int_{0}^{1} x \sin(\pi x) \, dx = \left[ x \left(-\frac{\cos(\pi x)}{\pi}\right) \right]_{0}^{1} - \int_{0}^{1} \left(-\frac{\cos(\pi x)}{\pi}\right) \, dx $$ $$ = \left[ -\frac{x \cos(\pi x)}{\pi} \right]_{0}^{1} + \frac{1}{\pi} \int_{0}^{1} \cos(\pi x) \, dx $$ Evaluate the first term: $$ \left[ -\frac{x \cos(\pi x)}{\pi} \right]_{0}^{1} = \left( -\frac{1 \cdot \cos(\pi)}{\pi} \right) - \left( -\frac{0 \cdot \cos(0)}{\pi} \right) $$ $$ = \left( -\frac{1 \cdot (-1)}{\pi} \right) - 0 = \frac{1}{\pi} $$ Evaluate the second term: $$ \frac{1}{\pi} \int_{0}^{1} \cos(\pi x) \, dx = \frac{1}{\pi} \left[ \frac{\sin(\pi x)}{\pi} \right]_{0}^{1} $$ $$ = \frac{1}{\pi} \left( \frac{\sin(\pi)}{\pi} - \frac{\sin(0)}{\pi} \right) = \frac{1}{\pi} \left( \frac{0}{\pi} - \frac{0}{\pi} \right) = 0 $$ Substitute these values back into the expression for $I$: $$ I = 2 \left( \frac{1}{\pi} + 0 \right) = \frac{2}{\pi} $$