Estimating the value of this Integral 
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:

*

*Black line $y=x$

*Curve bordering red area $x^{\sin x + \cos x}$

*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.
 A: 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 $432$.
