Q.If $\int_0^a \sqrt{x}\,dx = 2a\displaystyle\int_0^{\pi/2}\sin^3x\,dx$, then find the value of $\int_a^{a+1} x\,dx$

For the special family $\int_0^{\pi/2}\sin^n x\,dx$ and $\int_0^{\pi/2}\cos^n x\,dx$, a reduction formula gives the value directly from the integer $n$: multiply the chain of fractions $\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\frac{n-5}{n-4}\cdots$ down to $\frac23$ if $n$ is odd, or down to $\frac12$ times an extra factor of $\frac\pi2$ if $n$ is even. Because $\cos^n x$ on $[0,\pi/2]$ equals $\sin^n x$ reflected via $x\to\pi/2-x$ (Property V), the identical formula and the identical value of $n$ apply to both the sine and cosine versions — there is only one formula to remember. This is the fast route to any exercise integral that is a pure (or, with a small extension, mixed) power of $\sin x$ and $\cos x$ over a quarter-period interval, without repeating integration by parts from scratch each time. …