Integrals - Some Properties of Definite Integrals

Evaluate $I = \int_{0}^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx$.

Let $I = \int_{0}^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx \dots (1)$. Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$, we get: $I = \int_{0}^{\pi/2} \frac{\sin^{3/2} (\pi/2 - x)}{\sin^{3/2} (\pi/2 - x) + \cos^{3/2} (\pi/2 - x)} dx$. Since $\sin(\pi/2 - x) = \cos x$ and $\cos(\pi/2 - x) = \sin x$, we have $I = \int_{0}^{\pi/2} \frac{\cos^{3/2} x}{\cos^{3/2} x + \sin^{3/2} x} dx \dots (2)$. Adding (1) and (2): $2I = \int_{0}^{\pi/2} \frac{\sin^{3/2} x + \cos^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx = \int_{0}^{\pi/2} 1 dx$. Thus $2I = [x]_{0}^{\pi/2} = \pi/2$, which gives $I = \pi/4$.

This example applies the 'King's Rule' ($P_4$). By replacing $x$ with $(\pi/2 - x)$, the denominator remains the same while the numerator switches from sine to cosine. Adding the two forms eliminates the variable terms.

Evaluate $\int_{2}^{5} |x-3| dx$.

The function $|x-3|$ changes definition at $x=3$. Within $[2, 5]$, we split the integral at $x=3$: $\int_{2}^{5} |x-3| dx = \int_{2}^{3} |x-3| dx + \int_{3}^{5} |x-3| dx$. For $x \in [2, 3]$, $|x-3| = -(x-3) = 3-x$. For $x \in [3, 5]$, $|x-3| = x-3$. Therefore: $I = \int_{2}^{3} (3-x) dx + \int_{3}^{5} (x-3) dx$. $I = [3x - \frac{x^2}{2}]_{2}^{3} + [\frac{x^2}{2} - 3x]_{3}^{5}$. $I = (9 - 4.5) - (6 - 2) + (12.5 - 15) - (4.5 - 9) = 0.5 + 2 = 2.5$.

This demonstrates the splitting property ($P_2$). Since the modulus function is piecewise, the integral must be broken down based on the sign of the expression inside the absolute value.

Evaluate $\int_{-\pi/2}^{\pi/2} \sin^7 x dx$.

Let $f(x) = \sin^7 x$. Then $f(-x) = \sin^7(-x) = (-\sin x)^7 = -\sin^7 x$. Since $f(-x) = -f(x)$, $f(x)$ is an odd function. Using the property $\int_{-a}^{a} f(x) dx = 0$ for odd functions, we get $\int_{-\pi/2}^{\pi/2} \sin^7 x dx = 0$.

This uses property $P_7$. Recognizing that a function is odd on a symmetric interval $[-a, a]$ allows for an immediate solution without performing complex integration.