Definite Integration
Definite Integration
nta_pyq_2025_apr
Grade 12
Question:
If $I(m,n) = \displaystyle\int_0^1 x^{m-1}(1-x)^{n-1}\,dx$, $m,n>0$, then $I(9,14)+I(10,13)$ is
$I(19,27)$
$I(9,1)$
$I(1,13)$
$I(9,13)$
Step-by-Step Solution
Key Concept: Use the substitution $x = \sin^2\theta$ to write $I(m,n)$ as a trigonometric integral, then combine $I(9,14)+I(10,13)$ by factoring out common powers and using the Pythagorean identity $\sin^2\theta+\cos^2\theta=1$.
With $x = \sin^2\theta$:
$$I(m,n) = 2\int_0^{\pi/2}(\sin\theta)^{2m-1}(\cos\theta)^{2n-1}d\theta.$$
$$I(9,14)+I(10,13) = 2\int_0^{\pi/2}(\sin\theta)^{17}(\cos\theta)^{27}d\theta + 2\int_0^{\pi/2}(\sin\theta)^{19}(\cos\theta)^{25}d\theta$$
$$= 2\int_0^{\pi/2}(\sin\theta)^{17}(\cos\theta)^{25}\left[(\cos\theta)^2+(\sin\theta)^2\right]d\theta = 2\int_0^{\pi/2}(\sin\theta)^{17}(\cos\theta)^{25}d\theta = I(9,13).$$
Correct Answer: 4