Estimating the value of this Integral

Question

If I=$\int_0^1x^{sinx+cosx}dx$ then find the value of $[10I]$ where $[.]$ represent greatest integer function.

A)3

$\boxed{B)4}$

C)5

D)6

Mathongo, JEE sample questions

Method 1: I took a few points on the given curve and estimated out the area to be slightly less than 0.5 in the given interval, thus yielding the correct answer: 4.

Legend:

1. Black line $y=x$
2. Curve bordering red area $x^{\sin x + \cos x}$
3. Bounds as given.

Here is the attached Desmos link to depict the same

Method 2 (prompted):

$\hspace{42px} 1\leq sinx+cosx \leq \sqrt{2}$

$\implies x \geq x^{sinx+cosx} \geq x^{\sqrt{2}}$

$\implies \int_0^1 xdx \geq \int_0^1x^{sinx+cosx}dx \geq \int_0^1x^{\sqrt{2}}dx$

$\implies 0.5 \geq I \geq 0.414$

Thus, $\boxed{[10I]=4}$.

Would be glad if the community could come up with more alternative approaches.

Answer 1

Your solution is nice and simple.

On purpose, I shall make a complex one.

Consider $$y=x^{\sin (x)+\cos (x)}\quad \implies \quad \log(y)=[\sin (x)+\cos (x)]\log(x)$$ that is to say $$\log(y)=\log(x) \sum_{n=0}^\infty \frac{\sin \left(\frac{\pi n}{2}\right)+\cos \left(\frac{\pi n}{2}\right)}{n!} x^n$$ Truncate to some order $m$ and use $$y=e^{\log(y)}$$ This will give $$y=\sum_{n=1}^m \frac{P_n} {(n-1)!} x^n$$ where the $P_n$ are polynomials of degree $(n-1)$ in $\log(x)$ with no constant term. Using $L=\log(x)$, the first polynomials are $$\left( \begin{array}{cc} n & P_n \\ 1 & 1 \\ 2 & L \\ 3 & L^2-L \\ 4 & L^3-3 L^2-L \\ 5 & L^4-6 L^3-L^2+L \end{array} \right)$$

Now, we shall use the fact that, for positive $(p,q)$ $$\int_0^1 x^p \,\log ^q(x)\,dx=(-1)^q \frac {q!}{(p+1)^{q+1} }$$ Using all the above, we can now compute exactly the result for the successive values of $m$ $$\left( \begin{array}{cc} m & \text{result} \\ 1 & \frac{1}{2} \\ 2 & \frac{7}{18} \\ 3 & \frac{251}{576} \\ 4 & \frac{155819}{360000} \\ 5 & \frac{4204613}{9720000} \end{array} \right)$$

As soon as $m\geq 3$, you have the result.

If the problem was $\lfloor 100 I\rfloor$, as soon as $m\geq 3$, you would have $43$. If it was $\lfloor 1000 I\rfloor$, as soon as $m\geq 4$, you would have $432$.