Definite Integration – Properties of Definite Integrals – JEE Main 22 Jan 2026 Shift 2

Solution:

First, we evaluate $\alpha$. The integral can be split into two parts:
$$ \alpha = \int_{0}^{64} x^{1/3} dx – \int_{0}^{64} [x^{1/3}] dx $$
Evaluating the first part:
$$ \int_{0}^{64} x^{1/3} dx = \left[ \frac{3}{4} x^{4/3} \right]_{0}^{64} = \frac{3}{4} (64)^{4/3} = \frac{3}{4} (4^{3})^{4/3} = \frac{3}{4} (256) = 192 $$
Evaluating the second part involving the Greatest Integer Function. Let $t = x^{1/3}$, so limits are $0$ to $4$. The integer values change at $x = 1^3, 2^3, 3^3$.
$$ \int_{0}^{64} [x^{1/3}] dx = \int_{0}^{1} 0 dx + \int_{1}^{8} 1 dx + \int_{8}^{27} 2 dx + \int_{27}^{64} 3 dx $$
$$ = 0 + 1(8-1) + 2(27-8) + 3(64-27) $$
$$ = 7 + 2(19) + 3(37) $$
$$ = 7 + 38 + 111 = 156 $$
So, $\alpha = 192 – 156 = 36$.

Now we need to evaluate the second integral $I$:
$$ I = \frac{1}{\pi} \int_{0}^{36\pi} \frac{\sin^{2}\theta}{\sin^{6}\theta+\cos^{6}\theta} d\theta $$
The integrand $f(\theta) = \frac{\sin^{2}\theta}{\sin^{6}\theta+\cos^{6}\theta}$ has a period of $\frac{\pi}{2}$.
Using the property $\int_{0}^{nT} f(x) dx = n \int_{0}^{T} f(x) dx$:
The limit $36\pi$ contains $72$ periods of $\frac{\pi}{2}$.
$$ I = \frac{1}{\pi} \times 72 \int_{0}^{\pi/2} \frac{\sin^{2}\theta}{\sin^{6}\theta+\cos^{6}\theta} d\theta $$
Let $J = \int_{0}^{\pi/2} \frac{\sin^{2}\theta}{\sin^{6}\theta+\cos^{6}\theta} d\theta$.
Using King’s Property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$:
$$ J = \int_{0}^{\pi/2} \frac{\cos^{2}\theta}{\cos^{6}\theta+\sin^{6}\theta} d\theta $$
Adding the two expressions for $J$:
$$ 2J = \int_{0}^{\pi/2} \frac{\sin^{2}\theta + \cos^{2}\theta}{\sin^{6}\theta+\cos^{6}\theta} d\theta = \int_{0}^{\pi/2} \frac{1}{\sin^{6}\theta+\cos^{6}\theta} d\theta $$
We know that $\sin^{6}\theta+\cos^{6}\theta = 1 – 3\sin^{2}\theta\cos^{2}\theta$.
$$ 2J = \int_{0}^{\pi/2} \frac{1}{1 – \frac{3}{4}\sin^{2}(2\theta)} d\theta = \pi $$
$$ J = \frac{\pi}{2} $$
Substituting $J$ back into the expression for $I$:
$$ I = \frac{72}{\pi} \times \frac{\pi}{2} = 36 $$

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