Definite Integration

$\int_0^{\pi / 2} \frac{200 \sin x+100 \cos x}{\sin x+\cos x} d x$ is equal to

$\int_0^{\frac{\pi}{2}} \frac{\cos x d x}{\sqrt{1+\cos x \sin x}}=$

If $\mathrm{m} \in \mathrm{Z}^{+}, \mathrm{n}=2 \mathrm{~m}$ and $\int_0^{\frac{\pi}{2}} \sin ^{\mathrm{m}} \mathrm{x} \cos ^{\mathrm{n}} \mathrm{x} d \mathrm{x}=\mathrm{K}(\mathrm{m})$ $\int_0^{\frac{\pi}{2}} \sin ^m x…

$\int_0^3 \frac{3 x+1}{x^2+9} d x$ is equal to :

If $S_n=\int_0^{\frac{\pi}{2}} \frac{\sin (2 n-1) x}{\sin x} d x$ and $n$ is an integer, then $\mathrm{S}_{\mathrm{n}+1}-\mathrm{S}_{\mathrm{n}}=$

$f(x)=\left\{\begin{array}{l}x^2, 0 \leq x < 1 \\ \sqrt{x}, 1 \leq x \leq 2\end{array} \Rightarrow \int_0^2 f(x) d x=\right.$

$\lim _{n \rightarrow \infty} \frac{1}{\sqrt{n}}\left[1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\ldots+\frac{1}{\sqrt{n}}\right]=$

$\lim _{n \rightarrow \infty}\left[\frac{n+1}{n^2+1^2}+\frac{n+2}{n^2+2^2}+\frac{n+3}{n^2+3^2}+\ldots+\frac{n+2 n}{n^2+4 n^2}\right]=$

$\lim _{n \rightarrow \infty} n^4\left[\frac{1}{n^5}+\frac{1}{\left(n^2+1\right)^{\frac{5}{2}}}+\frac{1}{\left(n^2+4\right)^{\frac{5}{2}}}+\frac{1}{\left(n^2+9\right)^{\frac{5}{2}}}+\ldots+\right]=$

If $a \in \mathbb{Z}^{+},[x]$ is greatest integer not more than $x$ and $\int_0^a 2^{[x]} d x=127$, then $a=$

$$ \int_0^{50 \pi} \sqrt{1-\cos 2 x} d x= $$

$\int_{-\pi / 4}^{\pi / 3}\left|\tan \left(x-\frac{\pi}{6}\right)\right| d x=$