Q.By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$

Concept: Property of Symmetry — using the substitution $x \to \frac{\pi}{2} - x$ to exploit the complementary relationship between $\sin x$ and $\cos x$.

Let

Step 1: Substitute $x = \frac{\pi}{2} - t$, so $dx = -dt$. When $x=0$, $t=\frac{\pi}{2}$; when $x=\frac{\pi}{2}$, $t=0$.

Step 2: Renaming the dummy variable $t$ back to $x$, we have

Step 3: Add the two expressions for $I$:

Thus $I = \frac{\pi}{4}$.

The trick here is symmetry. When you see an integral from $0$ to $\pi/2$ with a ratio of sines and cosines, the substitution $x \to \pi/2 - x$ often turns the denominator into a mirror image of itself. This lets you add the original and transformed integrals, giving a simple result.

Let’s work through it.

1. Define the integral. Let

2. Apply the symmetry substitution. Use the property $\int_{0}^{a} f(x)\,dx = \int_{0}^{a} f(a-x)\,dx$. Here $a = \pi/2$, so replace $x$ by $\pi/2 - x$:

Recall the co-function identities:

$\sin(\pi/2 - x) = \cos x$ and $\cos(\pi/2 - x) = \sin x$.

So the integral becomes

3. Add the two forms. Now we have two expressions for $I$:

and

Add them:

The integrand simplifies to $1$ (provided the denominator is never zero on $[0, \pi/2]$, which it isn’t — both terms are non-negative and only vanish at the endpoints, but the sum is positive in between).

4. Evaluate the simple integral.

Hence $2I = \pi/2$, so

Method: The $\dfrac{f}{f+g}$ complementary-integral trick

For $\displaystyle\int_0^a \frac{f(x)}{f(x)+f(a-x)}\,dx$ (with $f,g$ swapping under $x\to a-x$), reflection produces the complementary fraction; the two add to $1$.

Steps

Step 1: Set $I$ and reflect with $x\to a-x$.

where the reflection swaps $f\leftrightarrow g$ (e.g. $\sqrt{\sin}\leftrightarrow\sqrt{\cos}$).

Step 2: Add the two forms.

The denominators are identical, so

Step 3: Solve for $I$.

This pattern always gives half the interval length, independent of the specific $f$.

Common Mistakes

Mistake 1: Thinking the exponent $\tfrac32$ changes the answer.

Why it's wrong: the $\frac{f}{f+g}$ reflection trick gives $\tfrac{\pi}{4}$ for any power on $\sin$/$\cos$, since the two complementary fractions always sum to $1$. Correct approach: reflect $x\to\tfrac{\pi}{2}-x$ and add — the exponent is irrelevant.

Mistake 2: Sign or limit errors when substituting $x=\tfrac{\pi}{2}-t$.

Why it's wrong: the $-dt$ and swapped limits must cancel; mishandling them corrupts the reflected integral. Correct approach: reverse the limits to absorb the minus sign, then rename $t$ back to $x$.