Nice problem.

Calculus Level 3

$\large I_{n}= \int_{0}^{\frac{\pi}{2}} \sin^n(x)\cos^{n+2}(x) \ dx$

For $I_n$ as defined above, what is the value of $\dfrac{I_{99}}{I_{101}}$ ?

The answer is 4.04.

1 solution

Chew-Seong Cheong
Apr 30, 2017

$\begin{aligned} I_n & = \int_0^\frac \pi 2 \sin^n x \cos^{n+2} x \ dx \\ & = \frac 12 B \left(\frac {n+1}2, \frac {n+3}2 \right) & \small \color{#3D99F6} \text{where } B(m,n) \text{ is the beta function.} \\ & = \frac {\Gamma\left(\frac {n+1}2 \right)\Gamma\left(\frac {n+3}2 \right)}{2\Gamma\left(n+2 \right)} & \small \color{#3D99F6} \text{where } \Gamma (s) \text{ is the gamma function.} \end{aligned}$

$\begin{aligned} \implies \frac {I_{99}}{I_{101}} & = \frac {\Gamma(50)\Gamma(51)}{2\Gamma(101)} \cdot \frac {2\Gamma(103)}{\Gamma(51)\Gamma(52)} \\ & = \frac {\Gamma(50)\Gamma(103)}{\Gamma(52)\Gamma(101)} \\ & = \frac {49!102!}{51!100!} \\ & = \frac {101\cdot 102}{50\cdot 51} \\ & = \boxed{4.04} \end{aligned}$

References:

Beta function
Gamma function

Nice solution. Did the same thing.

Sahil Silare - 4 years, 1 month ago

I think then you should learn integration!

Sahil Silare - 1 month, 2 weeks ago

I am not the age to learn that.

I learn these at school recently.

$( a \pm b ) ^ { 2 } = a ^ { 2 } \pm 2ab + b ^ { 2 }$ .

$( a + b ) ( a - b ) = a ^ { 2 } - b ^ { 2 }$ .

$\cdots$ .

Those are all polynomial expansion and factoring.

. . - 1 month, 2 weeks ago

@Sahil Silare This problem is useless if there is a person who does not know the way to integral like me.

I dont know how to integrate and what integral means.

And there are too many people like me.

. . - 2 months, 4 weeks ago

Nice problem.

The answer is 4.04.

1 solution

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