Integral Calculus
Beta/Gamma reduction; evaluate $\int_0^1 x^6(x^3-1)^{2022}dx$
MJMT_Full_Test_11
Grade 12
Question:
Value of $\displaystyle\int_0^1 x^6(x^3-1)^{2022}\,dx$ is
$\dfrac{3^{2022}(2022)!}{10\cdot13\cdot16\cdots6073}$
$\dfrac{3^{2022}(2022)!}{7\cdot10\cdot13\cdots6073}$
$\dfrac{3^{2022}(2022)!}{4\cdot7\cdot10\cdots6073}$
none of these
Step-by-Step Solution
Key Concept: Using reduction formula for $I_n=\int_0^1 x^6(x^3-1)^n dx$: integrate by parts, derive $I_n=\frac{3n}{3n+7}I_{n-1}$. Unwind to get product formula. Result involves $3^{2022}(2022)!/\text{product}$.
$\dfrac{3^{2022}(2022)!}{7\cdot10\cdot13\cdots6073}$.
Correct Answer: 2