JEE Main 2024 — Definite Integration Question with Solution

\begin{aligned} & I(21)=\int_\limits0^1\left(1-x^k\right)^{21} d x \\ & =\int_\limits0^1\left(1-x^k\right)\left(1-x^k\right)^{20} d x \\ & =\int_\limits0^1\left(1-x^k\right)^{20} d x-\int_0 x^k\left(1-x^k\right)^{20} d x \\ & I(21)=I(20)-\int_\limits0^1 x^k\left(1-x^k\right)^{20} d x \end{aligned}

I(21)=I(20)-\left\lfloor\frac{\left(1-x^k\right)^{21}}{-21 k} x-\int_\limits0^1 \frac{(1-x^k)^{21}}{-21 k} d x\right\rfloor

$$\begin{array}{rl}& I(21)=I(20)-\frac{1}{21k}I(20)\\ & \Rightarrow [I(21)](21k+1)=21KI(20)\\ & \Rightarrow 21K=147\Rightarrow K=7\end{array}$$